We will shortly turn to the complex-analytic approach to multiplicative number theory, which relies on the basic properties of complex analytic functions. In this supplement to the main notes, we quickly review the portions of complex analysis that we will be using in this course. We will not attempt a comprehensive review of this subject; for instance, we will completely neglect the conformal geometry or Riemann surface aspect of complex analysis, and we will also avoid using the various boundary convergence theorems for Taylor series or Dirichlet series (the latter type of result is traditionally utilised in multiplicative number theory, but I personally find them a little unintuitive to use, and will instead rely on a slightly different set of complex-analytic tools). We will also focus on the “local” structure of complex analytic functions, in particular adopting the philosophy that such functions behave locally like complex polynomials; the classical “global” theory of entire functions, while traditionally used in the theory of the Riemann zeta function, will be downplayed in these notes. On the other hand, we will play up the relationship between complex analysis and Fourier analysis, as we will incline to using the latter tool over the former in some of the subsequent material. (In the traditional approach to the subject, the Mellin transform is used in place of the Fourier transform, but we will not emphasise the role of the Mellin transform here.)

We begin by recalling the notion of a holomorphic function, which will later be shown to be essentially synonymous with that of a complex analytic function.

Definition 1 (Holomorphic function) Let {\Omega} be an open subset of {{\bf C}}, and let {f: \Omega \rightarrow {\bf C}} be a function. If {z \in {\bf C}}, we say that {f} is complex differentiable at {z} if the limit

\displaystyle  f'(z) := \lim_{h \rightarrow 0; h \in {\bf C} \backslash \{0\}} \frac{f(z+h)-f(z)}{h}

exists, in which case we refer to {f'(z)} as the (complex) derivative of {f} at {z}. If {f} is differentiable at every point {z} of {\Omega}, and the derivative {f': \Omega \rightarrow {\bf C}} is continuous, we say that {f} is holomorphic on {\Omega}.

Exercise 2 Show that a function {f: \Omega \rightarrow {\bf C}} is holomorphic if and only if the two-variable function {(x,y) \mapsto f(x+iy)} is continuously differentiable on {\{ (x,y) \in {\bf R}^2: x+iy \in \Omega\}} and obeys the Cauchy-Riemann equation

\displaystyle  \frac{\partial}{\partial x} f(x+iy) = \frac{1}{i} \frac{\partial}{\partial y} f(x+iy). \ \ \ \ \ (1)

Basic examples of holomorphic functions include complex polynomials

\displaystyle  P(z) = a_n z^n + \dots + a_1 z + a_0

as well as the complex exponential function

\displaystyle  \exp(z) := \sum_{n=0}^\infty \frac{z^n}{n!}

which are holomorphic on the entire complex plane {{\bf C}} (i.e., they are entire functions). The sum or product of two holomorphic functions is again holomorphic; the quotient of two holomorphic functions is holomorphic so long as the denominator is non-zero. Finally, the composition of two holomorphic functions is holomorphic wherever the composition is defined.

Exercise 3

  • (i) Establish Euler’s formula

    \displaystyle  \exp(x+iy) = e^x (\cos y + i \sin y)

    for all {x,y \in {\bf R}}. (Hint: it is a bit tricky to do this starting from the trigonometric definitions of sine and cosine; I recommend either using the Taylor series formulations of these functions instead, or alternatively relying on the ordinary differential equations obeyed by sine and cosine.)

  • (ii) Show that every non-zero complex number {z} has a complex logarithm {\log(z)} such that {\exp(\log(z))=z}, and that this logarithm is unique up to integer multiples of {2\pi i}.
  • (iii) Show that there exists a unique principal branch {\hbox{Log}(z)} of the complex logarithm in the region {{\bf C} \backslash (-\infty,0]}, defined by requiring {\hbox{Log}(z)} to be a logarithm of {z} with imaginary part between {-\pi} and {\pi}. Show that this principal branch is holomorphic with derivative {1/z}.

In real analysis, we have the fundamental theorem of calculus, which asserts that

\displaystyle  \int_a^b F'(t)\ dt = F(b) - F(a)

whenever {[a,b]} is a real interval and {F: [a,b] \rightarrow {\bf R}} is a continuously differentiable function. The complex analogue of this fact is that

\displaystyle  \int_\gamma F'(z)\ dz = F(\gamma(1)) - F(\gamma(0)) \ \ \ \ \ (2)

whenever {F: \Omega \rightarrow {\bf C}} is a holomorphic function, and {\gamma: [0,1] \rightarrow \Omega} is a contour in {\Omega}, by which we mean a piecewise continuously differentiable function, and the contour integral {\int_\gamma f(z)\ dz} for a continuous function {f} is defined via change of variables as

\displaystyle  \int_\gamma f(z)\ dz := \int_0^1 f(\gamma(t)) \gamma'(t)\ dt.

The complex fundamental theorem of calculus (2) follows easily from the real fundamental theorem and the chain rule.

In real analysis, we have the rather trivial fact that the integral of a continuous function on a closed contour is always zero:

\displaystyle  \int_a^b f(t)\ dt + \int_b^a f(t)\ dt = 0.

In complex analysis, the analogous fact is significantly more powerful, and is known as Cauchy’s theorem:

Theorem 4 (Cauchy’s theorem) Let {f: \Omega \rightarrow {\bf C}} be a holomorphic function in a simply connected open set {\Omega}, and let {\gamma: [0,1] \rightarrow \Omega} be a closed contour in {\Omega} (thus {\gamma(1)=\gamma(0)}). Then {\int_\gamma f(z)\ dz = 0}.

Exercise 5 Use Stokes’ theorem to give a proof of Cauchy’s theorem.

A useful reformulation of Cauchy’s theorem is that of contour shifting: if {f: \Omega \rightarrow {\bf C}} is a holomorphic function on a open set {\Omega}, and {\gamma, \tilde \gamma} are two contours in an open set {\Omega} with {\gamma(0)=\tilde \gamma(0)} and {\gamma(1) = \tilde \gamma(1)}, such that {\gamma} can be continuously deformed into {\tilde \gamma}, then {\int_\gamma f(z)\ dz = \int_{\tilde \gamma} f(z)\ dz}. A basic application of contour shifting is the Cauchy integral formula:

Theorem 6 (Cauchy integral formula) Let {f: \Omega \rightarrow {\bf C}} be a holomorphic function in a simply connected open set {\Omega}, and let {\gamma: [0,1] \rightarrow \Omega} be a closed contour which is simple (thus {\gamma} does not traverse any point more than once, with the exception of the endpoint {\gamma(0)=\gamma(1)} that is traversed twice), and which encloses a bounded region {U} in the anticlockwise direction. Then for any {z_0 \in U}, one has

\displaystyle  \int_\gamma \frac{f(z)}{z-z_0}\ dz= 2\pi i f(z_0).

Proof: Let {\varepsilon > 0} be a sufficiently small quantity. By contour shifting, one can replace the contour {\gamma} by the sum (concatenation) of three contours: a contour {\rho} from {\gamma(0)} to {z_0+\varepsilon}, a contour {C_\varepsilon} traversing the circle {\{z: |z-z_0|=\varepsilon\}} once anticlockwise, and the reversal {-\rho} of the contour {\rho} that goes from {z_0+\varepsilon} to {\gamma_0}. The contributions of the contours {\rho, -\rho} cancel each other, thus

\displaystyle \int_\gamma \frac{f(z)}{z-z_0}\ dz = \int_{C_\varepsilon} \frac{f(z)}{z-z_0}\ dz.

By a change of variables, the right-hand side can be expanded as

\displaystyle  2\pi i \int_0^1 f(z_0 + \varepsilon e^{2\pi i t})\ dt.

Sending {\varepsilon \rightarrow 0}, we obtain the claim. \Box

The Cauchy integral formula has many consequences. Specialising to the case when {\gamma} traverses a circle {\{ z: |z-z_0|=r\}} around {z_0}, we conclude the mean value property

\displaystyle  f(z_0) = \int_0^1 f(z_0 + re^{2\pi i t})\ dt \ \ \ \ \ (3)

whenever {f} is holomorphic in a neighbourhood of the disk {\{ z: |z-z_0| \leq r \}}. In a similar spirit, we have the maximum principle for holomorphic functions:

Lemma 7 (Maximum principle) Let {\Omega} be a simply connected open set, and let {\gamma} be a simple closed contour in {\Omega} enclosing a bounded region {U} anti-clockwise. Let {f: \Omega \rightarrow {\bf C}} be a holomorphic function. If we have the bound {|f(z)| \leq M} for all {z} on the contour {\gamma}, then we also have the bound {|f(z_0)| \leq M} for all {z_0 \in U}.

Proof: We use an argument of Landau. Fix {z_0 \in U}. From the Cauchy integral formula and the triangle inequality we have the bound

\displaystyle  |f(z_0)| \leq C_{z_0,\gamma} M

for some constant {C_{z_0,\gamma} > 0} depending on {z_0} and {\gamma}. This ostensibly looks like a weaker bound than what we want, but we can miraculously make the constant {C_{z_0,\gamma}} disappear by the “tensor power trick“. Namely, observe that if {f} is a holomorphic function bounded in magnitude by {M} on {\gamma}, and {n} is a natural number, then {f^n} is a holomorphic function bounded in magnitude by {M^n} on {\gamma}. Applying the preceding argument with {f, M} replaced by {f^n, M^n} we conclude that

\displaystyle  |f(z_0)|^n \leq C_{z_0,\gamma} M^n

and hence

\displaystyle  |f(z_0)| \leq C_{z_0,\gamma}^{1/n} M.

Sending {n \rightarrow \infty}, we obtain the claim. \Box

Another basic application of the integral formula is

Corollary 8 Every holomorphic function {f: \Omega \rightarrow {\bf C}} is complex analytic, thus it has a convergent Taylor series around every point {z_0} in the domain. In particular, holomorphic functions are smooth, and the derivative of a holomorphic function is again holomorphic.

Conversely, it is easy to see that complex analytic functions are holomorphic. Thus, the terms “complex analytic” and “holomorphic” are synonymous, at least when working on open domains. (On a non-open set {\Omega}, saying that {f} is analytic on {\Omega} is equivalent to asserting that {f} extends to a holomorphic function of an open neighbourhood of {\Omega}.) This is in marked contrast to real analysis, in which a function can be continuously differentiable, or even smooth, without being real analytic.

Proof: By translation, we may suppose that {z_0=0}. Let {C_r} be a a contour traversing the circle {\{ z: |z|=r\}} that is contained in the domain {\Omega}, then by the Cauchy integral formula one has

\displaystyle  f(z) = \frac{1}{2\pi i} \int_{C_r} \frac{f(w)}{w-z}\ dw

for all {z} in the disk {\{ z: |z| < r \}}. As {f} is continuously differentiable (and hence continuous) on {C_r}, it is bounded. From the geometric series formula

\displaystyle  \frac{1}{w-z} = \frac{1}{w} + \frac{1}{w^2} z + \frac{1}{w^3} z^2 + \dots

and dominated convergence, we conclude that

\displaystyle  f(z) = \sum_{n=0}^\infty (\frac{1}{2\pi i} \int_{C_r} \frac{f(w)}{w^{n+1}}\ dw) z^n

with the right-hand side an absolutely convergent series for {|z| < r}, and the claim follows. \Box

Exercise 9 Establish the generalised Cauchy integral formulae

\displaystyle  f^{(k)}(z_0) = \frac{k!}{2\pi i} \int_\gamma \frac{f(z)}{(z-z_0)^{k+1}}\ dz

for any non-negative integer {k}, where {f^{(k)}} is the {k}-fold complex derivative of {f}.

This in turn leads to a converse to Cauchy’s theorem, known as Morera’s theorem:

Corollary 10 (Morera’s theorem) Let {f: \Omega \rightarrow {\bf C}} be a continuous function on an open set {\Omega} with the property that {\int_\gamma f(z)\ dz = 0} for all closed contours {\gamma: [0,1] \rightarrow \Omega}. Then {f} is holomorphic.

Proof: We can of course assume {\Omega} to be non-empty and connected (hence path-connected). Fix a point {z_0 \in \Omega}, and define a “primitive” {F: \Omega \rightarrow {\bf C}} of {f} by defining {F(z_1) = \int_\gamma f(z)\ dz}, with {\gamma: [0,1] \rightarrow \Omega} being any contour from {z_0} to {z_1} (this is well defined by hypothesis). By mimicking the proof of the real fundamental theorem of calculus, we see that {F} is holomorphic with {F'=f}, and the claim now follows from Corollary 8. \Box

An important consequence of Morera’s theorem for us is

Corollary 11 (Locally uniform limit of holomorphic functions is holomorphic) Let {f_n: \Omega \rightarrow {\bf C}} be holomorphic functions on an open set {\Omega} which converge locally uniformly to a function {f: \Omega \rightarrow {\bf C}}. Then {f} is also holomorphic on {\Omega}.

Proof: By working locally we may assume that {\Omega} is a ball, and in particular simply connected. By Cauchy’s theorem, {\int_\gamma f_n(z)\ dz = 0} for all closed contours {\gamma} in {\Omega}. By local uniform convergence, this implies that {\int_\gamma f(z)\ dz = 0} for all such contours, and the claim then follows from Morera’s theorem. \Box

Now we study the zeroes of complex analytic functions. If a complex analytic function {f} vanishes at a point {z_0}, but is not identically zero in a neighbourhood of that point, then by Taylor expansion we see that {f} factors in a sufficiently small neighbourhood of {z_0} as

\displaystyle  f(z) = (z-z_0)^n g(z_0) \ \ \ \ \ (4)

for some natural number {n} (which we call the order or multiplicity of the zero at {f}) and some function {g} that is complex analytic and non-zero near {z_0}; this generalises the factor theorem for polynomials. In particular, the zero {z_0} is isolated if {f} does not vanish identically near {z_0}. We conclude that if {\Omega} is connected and {f} vanishes on a neighbourhood of some point {z_0} in {\Omega}, then it must vanish on all of {\Omega} (since the maximal connected neighbourhood of {z_0} in {\Omega} on which {f} vanishes cannot have any boundary point in {\Omega}). This implies unique continuation of analytic functions: if two complex analytic functions on {\Omega} agree on a non-empty open set, then they agree everywhere. In particular, if a complex analytic function does not vanish everywhere, then all of its zeroes are isolated, so in particular it has only finitely many zeroes on any given compact set.

Recall that a rational function is a function {f} which is a quotient {g/h} of two polynomials (at least outside of the set where {h} vanishes). Analogously, let us define a meromorphic function on an open set {\Omega} to be a function {f: \Omega \backslash S \rightarrow {\bf C}} defined outside of a discrete subset {S} of {\Omega} (the singularities of {f}), which is locally the quotient {g/h} of holomorphic functions, in the sense that for every {z_0 \in \Omega}, one has {f=g/h} in a neighbourhood of {z_0} excluding {S}, with {g, h} holomorphic near {z_0} and with {h} non-vanishing outside of {S}. If {z_0 \in S} and {g} has a zero of equal or higher order than {h} at {z_0}, then the singularity is removable and one can extend the meromorphic function holomorphically across {z_0} (by the holomorphic factor theorem (4)); otherwise, the singularity is non-removable and is known as a pole, whose order is equal to the difference between the order of {h} and the order of {g} at {z_0}. (If one wished, one could extend meromorphic functions to the poles by embedding {{\bf C}} in the Riemann sphere {{\bf C} \cup \{\infty\}} and mapping each pole to {\infty}, but we will not do so here. One could also consider non-meromorphic functions with essential singularities at various points, but we will have no need to analyse such singularities in this course.) If the order of a pole or zero is one, we say that it is simple; if it is two, we say it is double; and so forth.

Exercise 12 Show that the space of meromorphic functions on a non-empty open set {\Omega}, quotiented by almost everywhere equivalence, forms a field.

By quotienting two Taylor series, we see that if a meromorphic function {f} has a pole of order {n} at some point {z_0}, then it has a Laurent expansion

\displaystyle  f = \sum_{m=-n}^\infty a_m (z-z_0)^m,

absolutely convergent in a neighbourhood of {z_0} excluding {z_0} itself, and with {a_{-n}} non-zero. The Laurent coefficient {a_{-1}} has a special significance, and is called the residue of the meromorphic function {f} at {z_0}, which we will denote as {\hbox{Res}(f;z_0)}. The importance of this coefficient comes from the following significant generalisation of the Cauchy integral formula, known as the residue theorem:

Exercise 13 (Residue theorem) Let {f} be a meromorphic function on a simply connected domain {\Omega}, and let {\gamma} be a closed contour in {\Omega} enclosing a bounded region {U} anticlockwise, and avoiding all the singularities of {f}. Show that

\displaystyle  \int_\gamma f(z)\ dz = 2\pi i \sum_\rho \hbox{Res}(f;\rho)

where {\rho} is summed over all the poles of {f} that lie in {U}.

The residue theorem is particularly useful when applied to logarithmic derivatives {f'/f} of meromorphic functions {f}, because the residue is of a specific form:

Exercise 14 Let {f} be a meromorphic function on an open set {\Omega} that does not vanish identically. Show that the only poles of {f'/f} are simple poles (poles of order {1}), occurring at the poles and zeroes of {f} (after all removable singularities have been removed). Furthermore, the residue of {f'/f} at a pole {z_0} is an integer, equal to the order of zero of {f} if {f} has a zero at {z_0}, or equal to negative the order of pole at {f} if {f} has a pole at {z_0}.

Remark 15 The fact that residues of logarithmic derivatives of meromorphic functions are automatically integers is a remarkable feature of the complex analytic approach to multiplicative number theory, which is difficult (though not entirely impossible) to duplicate in other approaches to the subject. Here is a sample application of this integrality, which is challenging to reproduce by non-complex-analytic means: if {f} is meromorphic near {z_0}, and one has the bound {|\frac{f'}{f}(z_0+t)| \leq \frac{0.9}{t} + O(1)} as {t \rightarrow 0^+}, then {\frac{f'}{f}} must in fact stay bounded near {z_0}, because the only integer of magnitude less than {0.9} is zero.

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Van Vu and I have just uploaded to the arXiv our paper “Random matrices have simple spectrum“. Recall that an {n \times n} Hermitian matrix is said to have simple eigenvalues if all of its {n} eigenvalues are distinct. This is a very typical property of matrices to have: for instance, as discussed in this previous post, in the space of all {n \times n} Hermitian matrices, the space of matrices without all eigenvalues simple has codimension three, and for real symmetric cases this space has codimension two. In particular, given any random matrix ensemble of Hermitian or real symmetric matrices with an absolutely continuous distribution, we conclude that random matrices drawn from this ensemble will almost surely have simple eigenvalues.

For discrete random matrix ensembles, though, the above argument breaks down, even though general universality heuristics predict that the statistics of discrete ensembles should behave similarly to those of continuous ensembles. A model case here is the adjacency matrix {M_n} of an Erdös-Rényi graph – a graph on {n} vertices in which any pair of vertices has an independent probability {p} of being in the graph. For the purposes of this paper one should view {p} as fixed, e.g. {p=1/2}, while {n} is an asymptotic parameter going to infinity. In this context, our main result is the following (answering a question of Babai):

Theorem 1 With probability {1-o(1)}, {M_n} has simple eigenvalues.

Our argument works for more general Wigner-type matrix ensembles, but for sake of illustration we will stick with the Erdös-Renyi case. Previous work on local universality for such matrix models (e.g. the work of Erdos, Knowles, Yau, and Yin) was able to show that any individual eigenvalue gap {\lambda_{i+1}(M)-\lambda_i(M)} did not vanish with probability {1-o(1)} (in fact {1-O(n^{-c})} for some absolute constant {c>0}), but because there are {n} different gaps that one has to simultaneously ensure to be non-zero, this did not give Theorem 1 as one is forced to apply the union bound.

Our argument in fact gives simplicity of the spectrum with probability {1-O(n^{-A})} for any fixed {A}; in a subsequent paper we also show that it gives a quantitative lower bound on the eigenvalue gaps (analogous to how many results on the singularity probability of random matrices can be upgraded to a bound on the least singular value).

The basic idea of argument can be sketched as follows. Suppose that {M_n} has a repeated eigenvalue {\lambda}. We split

\displaystyle M_n = \begin{pmatrix} M_{n-1} & X \\ X^T & 0 \end{pmatrix}

for a random {n-1 \times n-1} minor {M_{n-1}} and a random sign vector {X}; crucially, {X} and {M_{n-1}} are independent. If {M_n} has a repeated eigenvalue {\lambda}, then by the Cauchy interlacing law, {M_{n-1}} also has an eigenvalue {\lambda}. We now write down the eigenvector equation for {M_n} at {\lambda}:

\displaystyle \begin{pmatrix} M_{n-1} & X \\ X^T & 0 \end{pmatrix} \begin{pmatrix} v \\ a \end{pmatrix} = \lambda \begin{pmatrix} v \\ a \end{pmatrix}.

Extracting the top {n-1} coefficients, we obtain

\displaystyle (M_{n-1} - \lambda) v + a X = 0.

If we let {w} be the {\lambda}-eigenvector of {M_{n-1}}, then by taking inner products with {w} we conclude that

\displaystyle a (w \cdot X) = 0;

we typically expect {a} to be non-zero, in which case we arrive at

\displaystyle w \cdot X = 0.

In other words, in order for {M_n} to have a repeated eigenvalue, the top right column {X} of {M_n} has to be orthogonal to an eigenvector {w} of the minor {M_{n-1}}. Note that {X} and {w} are going to be independent (once we specify which eigenvector of {M_{n-1}} to take as {w}). On the other hand, thanks to inverse Littlewood-Offord theory (specifically, we use an inverse Littlewood-Offord theorem of Nguyen and Vu), we know that the vector {X} is unlikely to be orthogonal to any given vector {w} independent of {X}, unless the coefficients of {w} are extremely special (specifically, that most of them lie in a generalised arithmetic progression). The main remaining difficulty is then to show that eigenvectors of a random matrix are typically not of this special form, and this relies on a conditioning argument originally used by Komlós to bound the singularity probability of a random sign matrix. (Basically, if an eigenvector has this special form, then one can use a fraction of the rows and columns of the random matrix to determine the eigenvector completely, while still preserving enough randomness in the remaining portion of the matrix so that this vector will in fact not be an eigenvector with high probability.)

Analytic number theory is only one of many different approaches to number theory. Another important branch of the subject is algebraic number theory, which studies algebraic structures (e.g. groups, rings, and fields) of number-theoretic interest. With this perspective, the classical field of rationals {{\bf Q}}, and the classical ring of integers {{\bf Z}}, are placed inside the much larger field {\overline{{\bf Q}}} of algebraic numbers, and the much larger ring {{\mathcal A}} of algebraic integers, respectively. Recall that an algebraic number is a root of a polynomial with integer coefficients, and an algebraic integer is a root of a monic polynomial with integer coefficients; thus for instance {\sqrt{2}} is an algebraic integer (a root of {x^2-2}), while {\sqrt{2}/2} is merely an algebraic number (a root of {4x^2-2}). For the purposes of this post, we will adopt the concrete (but somewhat artificial) perspective of viewing algebraic numbers and integers as lying inside the complex numbers {{\bf C}}, thus {{\mathcal A} \subset \overline{{\bf Q}} \subset {\bf C}}. (From a modern algebraic perspective, it is better to think of {\overline{{\bf Q}}} as existing as an abstract field separate from {{\bf C}}, but which has a number of embeddings into {{\bf C}} (as well as into other fields, such as the completed p-adics {{\bf C}_p}), no one of which should be considered favoured over any other; cf. this mathOverflow post. But for the rudimentary algebraic number theory in this post, we will not need to work at this level of abstraction.) In particular, we identify the algebraic integer {\sqrt{-d}} with the complex number {\sqrt{d} i} for any natural number {d}.

Exercise 1 Show that the field of algebraic numbers {\overline{{\bf Q}}} is indeed a field, and that the ring of algebraic integers {{\mathcal A}} is indeed a ring, and is in fact an integral domain. Also, show that {{\bf Z} = {\mathcal A} \cap {\bf Q}}, that is to say the ordinary integers are precisely the algebraic integers that are also rational. Because of this, we will sometimes refer to elements of {{\bf Z}} as rational integers.

In practice, the field {\overline{{\bf Q}}} is too big to conveniently work with directly, having infinite dimension (as a vector space) over {{\bf Q}}. Thus, algebraic number theory generally restricts attention to intermediate fields {{\bf Q} \subset F \subset \overline{{\bf Q}}} between {{\bf Q}} and {\overline{{\bf Q}}}, which are of finite dimension over {{\bf Q}}; that is to say, finite degree extensions of {{\bf Q}}. Such fields are known as algebraic number fields, or number fields for short. Apart from {{\bf Q}} itself, the simplest examples of such number fields are the quadratic fields, which have dimension exactly two over {{\bf Q}}.

Exercise 2 Show that if {\alpha} is a rational number that is not a perfect square, then the field {{\bf Q}(\sqrt{\alpha})} generated by {{\bf Q}} and either of the square roots of {\alpha} is a quadratic field. Conversely, show that all quadratic fields arise in this fashion. (Hint: show that every element of a quadratic field is a root of a quadratic polynomial over the rationals.)

The ring of algebraic integers {{\mathcal A}} is similarly too large to conveniently work with directly, so in algebraic number theory one usually works with the rings {{\mathcal O}_F := {\mathcal A} \cap F} of algebraic integers inside a given number field {F}. One can (and does) study this situation in great generality, but for the purposes of this post we shall restrict attention to a simple but illustrative special case, namely the quadratic fields with a certain type of negative discriminant. (The positive discriminant case will be briefly discussed in Remark 42 below.)

Exercise 3 Let {d} be a square-free natural number with {d=1\ (4)} or {d=2\ (4)}. Show that the ring {{\mathcal O} = {\mathcal O}_{{\bf Q}(\sqrt{-d})}} of algebraic integers in {{\bf Q}(\sqrt{-d})} is given by

\displaystyle  {\mathcal O} = {\bf Z}[\sqrt{-d}] = \{ a + b \sqrt{-d}: a,b \in {\bf Z} \}.

If instead {d} is square-free with {d=3\ (4)}, show that the ring {{\mathcal O} = {\mathcal O}_{{\bf Q}(\sqrt{-d})}} is instead given by

\displaystyle  {\mathcal O} = {\bf Z}[\frac{1+\sqrt{-d}}{2}] = \{ a + b \frac{1+\sqrt{-d}}{2}: a,b \in {\bf Z} \}.

What happens if {d} is not square-free, or negative?

Remark 4 In the case {d=3\ (4)}, it may naively appear more natural to work with the ring {{\bf Z}[\sqrt{-d}]}, which is an index two subring of {{\mathcal O}}. However, because this ring only captures some of the algebraic integers in {{\bf Q}(\sqrt{-d})} rather than all of them, the algebraic properties of these rings are somewhat worse than those of {{\mathcal O}} (in particular, they generally fail to be Dedekind domains) and so are not convenient to work with in algebraic number theory.

We refer to fields of the form {{\bf Q}(\sqrt{-d})} for natural square-free numbers {d} as quadratic fields of negative discriminant, and similarly refer to {{\mathcal O}_{{\bf Q}(\sqrt{-d})}} as a ring of quadratic integers of negative discriminant. Quadratic fields and quadratic integers of positive discriminant are just as important to analytic number theory as their negative discriminant counterparts, but we will restrict attention to the latter here for simplicity of discussion.

Thus, for instance, when {d=1}, the ring of integers in {{\bf Q}(\sqrt{-1})} is the ring of Gaussian integers

\displaystyle  {\bf Z}[\sqrt{-1}] = \{ x + y \sqrt{-1}: x,y \in {\bf Z} \}

and when {d=3}, the ring of integers in {{\bf Q}(\sqrt{-3})} is the ring of Eisenstein integers

\displaystyle  {\bf Z}[\omega] := \{ x + y \omega: x,y \in {\bf Z} \}

where {\omega := e^{2\pi i /3}} is a cube root of unity.

As these examples illustrate, the additive structure of a ring {{\mathcal O} = {\mathcal O}_{{\bf Q}(\sqrt{-d})}} of quadratic integers is that of a two-dimensional lattice in {{\bf C}}, which is isomorphic as an additive group to {{\bf Z}^2}. Thus, from an additive viewpoint, one can view quadratic integers as “two-dimensional” analogues of rational integers. From a multiplicative viewpoint, however, the quadratic integers (and more generally, integers in a number field) behave very similarly to the rational integers (as opposed to being some sort of “higher-dimensional” version of such integers). Indeed, a large part of basic algebraic number theory is devoted to treating the multiplicative theory of integers in number fields in a unified fashion, that naturally generalises the classical multiplicative theory of the rational integers.

For instance, every rational integer {n \in {\bf Z}} has an absolute value {|n| \in {\bf N} \cup \{0\}}, with the multiplicativity property {|nm| = |n| |m|} for {n,m \in {\bf Z}}, and the positivity property {|n| > 0} for all {n \neq 0}. Among other things, the absolute value detects units: {|n| = 1} if and only if {n} is a unit in {{\bf Z}} (that is to say, it is multiplicatively invertible in {{\bf Z}}). Similarly, in any ring of quadratic integers {{\mathcal O} = {\mathcal O}_{{\bf Q}(\sqrt{-d})}} with negative discriminant, we can assign a norm {N(n) \in {\bf N} \cup \{0\}} to any quadratic integer {n \in {\mathcal O}_{{\bf Q}(\sqrt{-d})}} by the formula

\displaystyle  N(n) = n \overline{n}

where {\overline{n}} is the complex conjugate of {n}. (When working with other number fields than quadratic fields of negative discriminant, one instead defines {N(n)} to be the product of all the Galois conjugates of {n}.) Thus for instance, when {d=1,2\ (4)} one has

\displaystyle  N(x + y \sqrt{-d}) = x^2 + dy^2 \ \ \ \ \ (1)

and when {d=3\ (4)} one has

\displaystyle  N(x + y \frac{1+\sqrt{-d}}{2}) = x^2 + xy + \frac{d+1}{4} y^2. \ \ \ \ \ (2)

Analogously to the rational integers, we have the multiplicativity property {N(nm) = N(n) N(m)} for {n,m \in {\mathcal O}} and the positivity property {N(n) > 0} for {n \neq 0}, and the units in {{\mathcal O}} are precisely the elements of norm one.

Exercise 5 Establish the three claims of the previous paragraph. Conclude that the units (invertible elements) of {{\mathcal O}} consist of the four elements {\pm 1, \pm i} if {d=1}, the six elements {\pm 1, \pm \omega, \pm \omega^2} if {d=3}, and the two elements {\pm 1} if {d \neq 1,3}.

For the rational integers, we of course have the fundamental theorem of arithmetic, which asserts that every non-zero rational integer can be uniquely factored (up to permutation and units) as the product of irreducible integers, that is to say non-zero, non-unit integers that cannot be factored into the product of integers of strictly smaller norm. As it turns out, the same claim is true for a few additional rings of quadratic integers, such as the Gaussian integers and Eisenstein integers, but fails in general; for instance, in the ring {{\bf Z}[\sqrt{-5}]}, we have the famous counterexample

\displaystyle  6 = 2 \times 3 = (1+\sqrt{-5}) (1-\sqrt{-5})

that decomposes {6} non-uniquely into the product of irreducibles in {{\bf Z}[\sqrt{-5}]}. Nevertheless, it is an important fact that the fundamental theorem of arithmetic can be salvaged if one uses an “idealised” notion of a number in a ring of integers {{\mathcal O}}, now known in modern language as an ideal of that ring. For instance, in {{\bf Z}[\sqrt{-5}]}, the principal ideal {(6)} turns out to uniquely factor into the product of (non-principal) ideals {(2) + (1+\sqrt{-5}), (2) + (1-\sqrt{-5}), (3) + (1+\sqrt{-5}), (3) + (1-\sqrt{-5})}; see Exercise 27. We will review the basic theory of ideals in number fields (focusing primarily on quadratic fields of negative discriminant) below the fold.

The norm forms (1), (2) can be viewed as examples of positive definite quadratic forms {Q: {\bf Z}^2 \rightarrow {\bf Z}} over the integers, by which we mean a polynomial of the form

\displaystyle  Q(x,y) = ax^2 + bxy + cy^2

for some integer coefficients {a,b,c}. One can declare two quadratic forms {Q, Q': {\bf Z}^2 \rightarrow {\bf Z}} to be equivalent if one can transform one to the other by an invertible linear transformation {T: {\bf Z}^2 \rightarrow {\bf Z}^2}, so that {Q' = Q \circ T}. For example, the quadratic forms {(x,y) \mapsto x^2 + y^2} and {(x',y') \mapsto 2 (x')^2 + 2 x' y' + (y')^2} are equivalent, as can be seen by using the invertible linear transformation {(x,y) = (x',x'+y')}. Such equivalences correspond to the different choices of basis available when expressing a ring such as {{\mathcal O}} (or an ideal thereof) additively as a copy of {{\bf Z}^2}.

There is an important and classical invariant of a quadratic form {(x,y) \mapsto ax^2 + bxy + c y^2}, namely the discriminant {\Delta := b^2 - 4ac}, which will of course be familiar to most readers via the quadratic formula, which among other things tells us that a quadratic form will be positive definite precisely when its discriminant is negative. It is not difficult (particularly if one exploits the multiplicativity of the determinant of {2 \times 2} matrices) to show that two equivalent quadratic forms have the same discriminant. Thus for instance any quadratic form equivalent to (1) has discriminant {-4d}, while any quadratic form equivalent to (2) has discriminant {-d}. Thus we see that each ring {{\mathcal O}[\sqrt{-d}]} of quadratic integers is associated with a certain negative discriminant {D}, defined to equal {-4d} when {d=1,2\ (4)} and {-d} when {d=3\ (4)}.

Exercise 6 (Geometric interpretation of discriminant) Let {Q: {\bf Z}^2 \rightarrow {\bf Z}} be a quadratic form of negative discriminant {D}, and extend it to a real form {Q: {\bf R}^2 \rightarrow {\bf R}} in the obvious fashion. Show that for any {X>0}, the set {\{ (x,y) \in {\bf R}^2: Q(x,y) \leq X \}} is an ellipse of area {2\pi X / \sqrt{|D|}}.

It is natural to ask the converse question: if two quadratic forms have the same discriminant, are they necessarily equivalent? For certain choices of discriminant, this is the case:

Exercise 7 Show that any quadratic form {ax^2+bxy+cy^2} of discriminant {-4} is equivalent to the form {x^2+y^2}, and any quadratic form of discriminant {-3} is equivalent to {x^2+xy+y^2}. (Hint: use elementary transformations to try to make {|b|} as small as possible, to the point where one only has to check a finite number of cases; this argument is due to Legendre.) More generally, show that for any negative discriminant {D}, there are only finitely many quadratic forms of that discriminant up to equivalence (a result first established by Gauss).

Unfortunately, for most choices of discriminant, the converse question fails; for instance, the quadratic forms {x^2+5y^2} and {2x^2+2xy+3y^2} both have discriminant {-20}, but are not equivalent (Exercise 38). This particular failure of equivalence turns out to be intimately related to the failure of unique factorisation in the ring {{\bf Z}[\sqrt{-5}]}.

It turns out that there is a fundamental connection between quadratic fields, equivalence classes of quadratic forms of a given discriminant, and real Dirichlet characters, thus connecting the material discussed above with the last section of the previous set of notes. Here is a typical instance of this connection:

Proposition 8 Let {\chi_4: {\bf N} \rightarrow {\bf R}} be the real non-principal Dirichlet character of modulus {4}, or more explicitly {\chi_4(n)} is equal to {+1} when {n = 1\ (4)}, {-1} when {n = 3\ (4)}, and {0} when {n = 0,2\ (4)}.

  • (i) For any natural number {n}, the number of Gaussian integers {m \in {\bf Z}[\sqrt{-1}]} with norm {N(m)=n} is equal to {4(1 * \chi_4)(n)}. Equivalently, the number of solutions to the equation {n = x^2+y^2} with {x,y \in{\bf Z}} is {4(1*\chi_4)(n)}. (Here, as in the previous post, the symbol {*} denotes Dirichlet convolution.)
  • (ii) For any natural number {n}, the number of Gaussian integers {m \in {\bf Z}[\sqrt{-1}]} that divide {n} (thus {n = dm} for some {d \in {\bf Z}[\sqrt{-1}]}) is {4(1*1*1*\mu\chi_4)(n)}.

We will prove this proposition later in these notes. We observe that as a special case of part (i) of this proposition, we recover the Fermat two-square theorem: an odd prime {p} is expressible as the sum of two squares if and only if {p = 1\ (4)}. This proposition should also be compared with the fact, used crucially in the previous post to prove Dirichlet’s theorem, that {1*\chi(n)} is non-negative for any {n}, and at least one when {n} is a square, for any quadratic character {\chi}.

As an illustration of the relevance of such connections to analytic number theory, let us now explicitly compute {L(1,\chi_4)}.

Corollary 9 {L(1,\chi_4) = \frac{\pi}{4}}.

This particular identity is also known as the Leibniz formula.

Proof: For a large number {x}, consider the quantity

\displaystyle  \sum_{n \in {\bf Z}[\sqrt{-1}]: N(n) \leq x} 1

of all the Gaussian integers of norm less than {x}. On the one hand, this is the same as the number of lattice points of {{\bf Z}^2} in the disk {\{ (a,b) \in {\bf R}^2: a^2+b^2 \leq x \}} of radius {\sqrt{x}}. Placing a unit square centred at each such lattice point, we obtain a region which differs from the disk by a region contained in an annulus of area {O(\sqrt{x})}. As the area of the disk is {\pi x}, we conclude the Gauss bound

\displaystyle  \sum_{n \in {\bf Z}[\sqrt{-1}]: N(n) \leq x} 1 = \pi x + O(\sqrt{x}).

On the other hand, by Proposition 8(i) (and removing the {n=0} contribution), we see that

\displaystyle  \sum_{n \in {\bf Z}[\sqrt{-1}]: N(n) \leq x} 1 = 1 + 4 \sum_{n \leq x} 1 * \chi_4(n).

Now we use the Dirichlet hyperbola method to expand the right-hand side sum, first expressing

\displaystyle  \sum_{n \leq x} 1 * \chi_4(n) = \sum_{d \leq \sqrt{x}} \chi_4(d) \sum_{m \leq x/d} 1 + \sum_{m \leq \sqrt{x}} \sum_{d \leq x/m} \chi_4(d)

\displaystyle  - (\sum_{d \leq \sqrt{x}} \chi_4(d)) (\sum_{m \leq \sqrt{x}} 1)

and then using the bounds {\sum_{d \leq y} \chi_4(d) = O(1)}, {\sum_{m \leq y} 1 = y + O(1)}, {\sum_{d \leq \sqrt{x}} \frac{\chi_4(d)}{d} = L(1,\chi_4) + O(\frac{1}{\sqrt{x}})} from the previous set of notes to conclude that

\displaystyle  \sum_{n \leq x} 1 * \chi_4(n) = x L(1,\chi_4) + O(\sqrt{x}).

Comparing the two formulae for {\sum_{n \in {\bf Z}[\sqrt{-1}]: N(n) \leq x} 1} and sending {x \rightarrow \infty}, we obtain the claim. \Box

Exercise 10 Give an alternate proof of Corollary 9 that relies on obtaining asymptotics for the Dirichlet series {\sum_{n \in {\bf Z}} \frac{1 * \chi_4(n)}{n^s}} as {s \rightarrow 1^+}, rather than using the Dirichlet hyperbola method.

Exercise 11 Give a direct proof of Corollary 9 that does not use Proposition 8, instead using Taylor expansion of the complex logarithm {\log(1+z)}. (One can also use Taylor expansions of some other functions related to the complex logarithm here, such as the arctangent function.)

More generally, one can relate {L(1,\chi)} for a real Dirichlet character {\chi} with the number of inequivalent quadratic forms of a certain discriminant, via the famous class number formula; we will give a special case of this formula below the fold.

The material here is only a very rudimentary introduction to algebraic number theory, and is not essential to the rest of the course. A slightly expanded version of the material here, from the perspective of analytic number theory, may be found in Sections 5 and 6 of Davenport’s book. A more in-depth treatment of algebraic number theory may be found in a number of texts, e.g. Fröhlich and Taylor.

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In analytic number theory, an arithmetic function is simply a function {f: {\bf N} \rightarrow {\bf C}} from the natural numbers {{\bf N} = \{1,2,3,\dots\}} to the real or complex numbers. (One occasionally also considers arithmetic functions taking values in more general rings than {{\bf R}} or {{\bf C}}, as in this previous blog post, but we will restrict attention here to the classical situation of real or complex arithmetic functions.) Experience has shown that a particularly tractable and relevant class of arithmetic functions for analytic number theory are the multiplicative fun3ctions, which are arithmetic functions {f: {\bf N} \rightarrow {\bf C}} with the additional property that

\displaystyle f(nm) = f(n) f(m) \ \ \ \ \ (1)

 

whenever {n,m \in{\bf N}} are coprime. (One also considers arithmetic functions, such as the logarithm function {L(n) := \log n} or the von Mangoldt function, that are not genuinely multiplicative, but interact closely with multiplicative functions, and can be viewed as “derived” versions of multiplicative functions; see this previous post.) A typical example of a multiplicative function is the divisor function

\displaystyle \tau(n) := \sum_{d|n} 1 \ \ \ \ \ (2)

 

that counts the number of divisors of a natural number {n}. (The divisor function {n \mapsto \tau(n)} is also denoted {n \mapsto d(n)} in the literature.) The study of asymptotic behaviour of multiplicative functions (and their relatives) is known as multiplicative number theory, and is a basic cornerstone of modern analytic number theory.

There are various approaches to multiplicative number theory, each of which focuses on different asymptotic statistics of arithmetic functions {f}. In elementary multiplicative number theory, which is the focus of this set of notes, particular emphasis is given on the following two statistics of a given arithmetic function {f: {\bf N} \rightarrow {\bf C}}:

  1. The summatory functions

    \displaystyle \sum_{n \leq x} f(n)

    of an arithmetic function {f}, as well as the associated natural density

    \displaystyle \lim_{x \rightarrow \infty} \frac{1}{x} \sum_{n \leq x} f(n)

    (if it exists).

  2. The logarithmic sums

    \displaystyle \sum_{n\leq x} \frac{f(n)}{n}

    of an arithmetic function {f}, as well as the associated logarithmic density

    \displaystyle \lim_{x \rightarrow \infty} \frac{1}{\log x} \sum_{n \leq x} \frac{f(n)}{n}

    (if it exists).

Here, we are normalising the arithmetic function {f} being studied to be of roughly unit size up to logarithms, obeying bounds such as {f(n)=O(1)}, {f(n) = O(\log^{O(1)} n)}, or at worst

\displaystyle f(n) = O(n^{o(1)}). \ \ \ \ \ (3)

 

A classical case of interest is when {f} is an indicator function {f=1_A} of some set {A} of natural numbers, in which case we also refer to the natural or logarithmic density of {f} as the natural or logarithmic density of {A} respectively. However, in analytic number theory it is usually more convenient to replace such indicator functions with other related functions that have better multiplicative properties. For instance, the indicator function {1_{\mathcal P}} of the primes is often replaced with the von Mangoldt function {\Lambda}.

Typically, the logarithmic sums are relatively easy to control, but the summatory functions require more effort in order to obtain satisfactory estimates; see Exercise 7 below.

If an arithmetic function {f} is multiplicative (or closely related to a multiplicative function), then there is an important further statistic on an arithmetic function {f} beyond the summatory function and the logarithmic sum, namely the Dirichlet series

\displaystyle {\mathcal D}f(s) := \sum_{n=1}^\infty \frac{f(n)}{n^s} \ \ \ \ \ (4)

 

for various real or complex numbers {s}. Under the hypothesis (3), this series is absolutely convergent for real numbers {s>1}, or more generally for complex numbers {s} with {\hbox{Re}(s)>1}. As we will see below the fold, when {f} is multiplicative then the Dirichlet series enjoys an important Euler product factorisation which has many consequences for analytic number theory.

In the elementary approach to multiplicative number theory presented in this set of notes, we consider Dirichlet series only for real numbers {s>1} (and focusing particularly on the asymptotic behaviour as {s \rightarrow 1^+}); in later notes we will focus instead on the important complex-analytic approach to multiplicative number theory, in which the Dirichlet series (4) play a central role, and are defined not only for complex numbers with large real part, but are often extended analytically or meromorphically to the rest of the complex plane as well.

Remark 1 The elementary and complex-analytic approaches to multiplicative number theory are the two classical approaches to the subject. One could also consider a more “Fourier-analytic” approach, in which one studies convolution-type statistics such as

\displaystyle \sum_n \frac{f(n)}{n} G( t - \log n ) \ \ \ \ \ (5)

 

as {t \rightarrow \infty} for various cutoff functions {G: {\bf R} \rightarrow {\bf C}}, such as smooth, compactly supported functions. See for instance this previous blog post for an instance of such an approach. Another related approach is the “pretentious” approach to multiplicative number theory currently being developed by Granville-Soundararajan and their collaborators. We will occasionally make reference to these more modern approaches in these notes, but will primarily focus on the classical approaches.

To reverse the process and derive control on summatory functions or logarithmic sums starting from control of Dirichlet series is trickier, and usually requires one to allow {s} to be complex-valued rather than real-valued if one wants to obtain really accurate estimates; we will return to this point in subsequent notes. However, there is a cheap way to get upper bounds on such sums, known as Rankin’s trick, which we will discuss later in these notes.

The basic strategy of elementary multiplicative theory is to first gather useful estimates on the statistics of “smooth” or “non-oscillatory” functions, such as the constant function {n \mapsto 1}, the harmonic function {n \mapsto \frac{1}{n}}, or the logarithm function {n \mapsto \log n}; one also considers the statistics of periodic functions such as Dirichlet characters. These functions can be understood without any multiplicative number theory, using basic tools from real analysis such as the (quantitative version of the) integral test or summation by parts. Once one understands the statistics of these basic functions, one can then move on to statistics of more arithmetically interesting functions, such as the divisor function (2) or the von Mangoldt function {\Lambda} that we will discuss below. A key tool to relate these functions to each other is that of Dirichlet convolution, which is an operation that interacts well with summatory functions, logarithmic sums, and particularly well with Dirichlet series.

This is only an introduction to elementary multiplicative number theory techniques. More in-depth treatments may be found in this text of Montgomery-Vaughan, or this text of Bateman-Diamond.

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Many problems and results in analytic prime number theory can be formulated in the following general form: given a collection of (affine-)linear forms {L_1(n),\dots,L_k(n)}, none of which is a multiple of any other, find a number {n} such that a certain property {P( L_1(n),\dots,L_k(n) )} of the linear forms {L_1(n),\dots,L_k(n)} are true. For instance:

  • For the twin prime conjecture, one can use the linear forms {L_1(n) := n}, {L_2(n) := n+2}, and the property {P( L_1(n), L_2(n) )} in question is the assertion that {L_1(n)} and {L_2(n)} are both prime.
  • For the even Goldbach conjecture, the claim is similar but one uses the linear forms {L_1(n) := n}, {L_2(n) := N-n} for some even integer {N}.
  • For Chen’s theorem, we use the same linear forms {L_1(n),L_2(n)} as in the previous two cases, but now {P(L_1(n), L_2(n))} is the assertion that {L_1(n)} is prime and {L_2(n)} is an almost prime (in the sense that there are at most two prime factors).
  • In the recent results establishing bounded gaps between primes, we use the linear forms {L_i(n) = n + h_i} for some admissible tuple {h_1,\dots,h_k}, and take {P(L_1(n),\dots,L_k(n))} to be the assertion that at least two of {L_1(n),\dots,L_k(n)} are prime.

For these sorts of results, one can try a sieve-theoretic approach, which can broadly be formulated as follows:

  1. First, one chooses a carefully selected sieve weight {\nu: {\bf N} \rightarrow {\bf R}^+}, which could for instance be a non-negative function having a divisor sum form

    \displaystyle  \nu(n) := \sum_{d_1|L_1(n), \dots, d_k|L_k(n); d_1 \dots d_k \leq x^{1-\varepsilon}} \lambda_{d_1,\dots,d_k}

    for some coefficients {\lambda_{d_1,\dots,d_k}}, where {x} is a natural scale parameter. The precise choice of sieve weight is often quite a delicate matter, but will not be discussed here. (In some cases, one may work with multiple sieve weights {\nu_1, \nu_2, \dots}.)

  2. Next, one uses tools from analytic number theory (such as the Bombieri-Vinogradov theorem) to obtain upper and lower bounds for sums such as

    \displaystyle  \sum_n \nu(n) \ \ \ \ \ (1)

    or

    \displaystyle  \sum_n \nu(n) 1_{L_i(n) \hbox{ prime}} \ \ \ \ \ (2)

    or more generally of the form

    \displaystyle  \sum_n \nu(n) f(L_i(n)) \ \ \ \ \ (3)

    where {f(L_i(n))} is some “arithmetic” function involving the prime factorisation of {L_i(n)} (we will be a bit vague about what this means precisely, but a typical choice of {f} might be a Dirichlet convolution {\alpha*\beta(L_i(n))} of two other arithmetic functions {\alpha,\beta}).

  3. Using some combinatorial arguments, one manipulates these upper and lower bounds, together with the non-negative nature of {\nu}, to conclude the existence of an {n} in the support of {\nu} (or of at least one of the sieve weights {\nu_1, \nu_2, \dots} being considered) for which {P( L_1(n), \dots, L_k(n) )} holds

For instance, in the recent results on bounded gaps between primes, one selects a sieve weight {\nu} for which one has upper bounds on

\displaystyle  \sum_n \nu(n)

and lower bounds on

\displaystyle  \sum_n \nu(n) 1_{n+h_i \hbox{ prime}}

so that one can show that the expression

\displaystyle  \sum_n \nu(n) (\sum_{i=1}^k 1_{n+h_i \hbox{ prime}} - 1)

is strictly positive, which implies the existence of an {n} in the support of {\nu} such that at least two of {n+h_1,\dots,n+h_k} are prime. As another example, to prove Chen’s theorem to find {n} such that {L_1(n)} is prime and {L_2(n)} is almost prime, one uses a variety of sieve weights to produce a lower bound for

\displaystyle  S_1 := \sum_{n \leq x} 1_{L_1(n) \hbox{ prime}} 1_{L_2(n) \hbox{ rough}}

and an upper bound for

\displaystyle  S_2 := \sum_{z \leq p < x^{1/3}} \sum_{n \leq x} 1_{L_1(n) \hbox{ prime}} 1_{p|L_2(n)} 1_{L_2(n) \hbox{ rough}}

and

\displaystyle  S_3 := \sum_{n \leq x} 1_{L_1(n) \hbox{ prime}} 1_{L_2(n)=pqr \hbox{ for some } z \leq p \leq x^{1/3} < q \leq r},

where {z} is some parameter between {1} and {x^{1/3}}, and “rough” means that all prime factors are at least {z}. One can observe that if {S_1 - \frac{1}{2} S_2 - \frac{1}{2} S_3 > 0}, then there must be at least one {n} for which {L_1(n)} is prime and {L_2(n)} is almost prime, since for any rough number {m}, the quantity

\displaystyle  1 - \frac{1}{2} \sum_{z \leq p < x^{1/3}} 1_{p|m} - \frac{1}{2} \sum_{z \leq p \leq x^{1/3} < q \leq r} 1_{m = pqr}

is only positive when {m} is an almost prime (if {m} has three or more factors, then either it has at least two factors less than {x^{1/3}}, or it is of the form {pqr} for some {p \leq x^{1/3} < q \leq r}). The upper and lower bounds on {S_1,S_2,S_3} are ultimately produced via asymptotics for expressions of the form (1), (2), (3) for various divisor sums {\nu} and various arithmetic functions {f}.

Unfortunately, there is an obstruction to sieve-theoretic techniques working for certain types of properties {P(L_1(n),\dots,L_k(n))}, which Zeb Brady and I recently formalised at an AIM workshop this week. To state the result, we recall the Liouville function {\lambda(n)}, defined by setting {\lambda(n) = (-1)^j} whenever {n} is the product of exactly {j} primes (counting multiplicity). Define a sign pattern to be an element {(\epsilon_1,\dots,\epsilon_k)} of the discrete cube {\{-1,+1\}^k}. Given a property {P(l_1,\dots,l_k)} of {k} natural numbers {l_1,\dots,l_k}, we say that a sign pattern {(\epsilon_1,\dots,\epsilon_k)} is forbidden by {P} if there does not exist any natural numbers {l_1,\dots,l_k} obeying {P(l_1,\dots,l_k)} for which

\displaystyle  (\lambda(l_1),\dots,\lambda(l_k)) = (\epsilon_1,\dots,\epsilon_k).

Example 1 Let {P(l_1,l_2,l_3)} be the property that at least two of {l_1,l_2,l_3} are prime. Then the sign patterns {(+1,+1,+1)}, {(+1,+1,-1)}, {(+1,-1,+1)}, {(-1,+1,+1)} are forbidden, because prime numbers have a Liouville function of {-1}, so that {P(l_1,l_2,l_3)} can only occur when at least two of {\lambda(l_1),\lambda(l_2), \lambda(l_3)} are equal to {-1}.

Example 2 Let {P(l_1,l_2)} be the property that {l_1} is prime and {l_2} is almost prime. Then the only forbidden sign patterns are {(+1,+1)} and {(+1,-1)}.

Example 3 Let {P(l_1,l_2)} be the property that {l_1} and {l_2} are both prime. Then {(+1,+1), (+1,-1), (-1,+1)} are all forbidden sign patterns.

We then have a parity obstruction as soon as {P} has “too many” forbidden sign patterns, in the following (slightly informal) sense:

Claim 1 (Parity obstruction) Suppose {P(l_1,\dots,l_k)} is such that that the convex hull of the forbidden sign patterns of {P} contains the origin. Then one cannot use the above sieve-theoretic approach to establish the existence of an {n} such that {P(L_1(n),\dots,L_k(n))} holds.

Thus for instance, the property in Example 3 is subject to the parity obstruction since {0} is a convex combination of {(+1,-1)} and {(-1,+1)}, whereas the properties in Examples 1, 2 are not. One can also check that the property “at least {j} of the {k} numbers {l_1,\dots,l_k} is prime” is subject to the parity obstruction as soon as {j \geq \frac{k}{2}+1}. Thus, the largest number of elements of a {k}-tuple that one can force to be prime by purely sieve-theoretic methods is {k/2}, rounded up.

This claim is not precisely a theorem, because it presumes a certain “Liouville pseudorandomness conjecture” (a very close cousin of the more well known “Möbius pseudorandomness conjecture”) which is a bit difficult to formalise precisely. However, this conjecture is widely believed by analytic number theorists, see e.g. this blog post for a discussion. (Note though that there are scenarios, most notably the “Siegel zero” scenario, in which there is a severe breakdown of this pseudorandomness conjecture, and the parity obstruction then disappears. A typical instance of this is Heath-Brown’s proof of the twin prime conjecture (which would ordinarily be subject to the parity obstruction) under the hypothesis of a Siegel zero.) The obstruction also does not prevent the establishment of an {n} such that {P(L_1(n),\dots,L_k(n))} holds by introducing additional sieve axioms beyond upper and lower bounds on quantities such as (1), (2), (3). The proof of the Friedlander-Iwaniec theorem is a good example of this latter scenario.

Now we give a (slightly nonrigorous) proof of the claim.

Proof: (Nonrigorous) Suppose that the convex hull of the forbidden sign patterns contain the origin. Then we can find non-negative numbers {p_{\epsilon_1,\dots,\epsilon_k}} for sign patterns {(\epsilon_1,\dots,\epsilon_k)}, which sum to {1}, are non-zero only for forbidden sign patterns, and which have mean zero in the sense that

\displaystyle  \sum_{(\epsilon_1,\dots,\epsilon_k)} p_{\epsilon_1,\dots,\epsilon_k} \epsilon_i = 0

for all {i=1,\dots,k}. By Fourier expansion (or Lagrange interpolation), one can then write {p_{\epsilon_1,\dots,\epsilon_k}} as a polynomial

\displaystyle  p_{\epsilon_1,\dots,\epsilon_k} = 1 + Q( \epsilon_1,\dots,\epsilon_k)

where {Q(t_1,\dots,t_k)} is a polynomial in {k} variables that is a linear combination of monomials {t_{i_1} \dots t_{i_r}} with {i_1 < \dots < i_r} and {r \geq 2} (thus {Q} has no constant or linear terms, and no monomials with repeated terms). The point is that the mean zero condition allows one to eliminate the linear terms. If we now consider the weight function

\displaystyle  w(n) := 1 + Q( \lambda(L_1(n)), \dots, \lambda(L_k(n)) )

then {w} is non-negative, is supported solely on {n} for which {(\lambda(L_1(n)),\dots,\lambda(L_k(n)))} is a forbidden pattern, and is equal to {1} plus a linear combination of monomials {\lambda(L_{i_1}(n)) \dots \lambda(L_{i_r}(n))} with {r \geq 2}.

The Liouville pseudorandomness principle then predicts that sums of the form

\displaystyle  \sum_n \nu(n) Q( \lambda(L_1(n)), \dots, \lambda(L_k(n)) )

and

\displaystyle  \sum_n \nu(n) Q( \lambda(L_1(n)), \dots, \lambda(L_k(n)) ) 1_{L_i(n) \hbox{ prime}}

or more generally

\displaystyle  \sum_n \nu(n) Q( \lambda(L_1(n)), \dots, \lambda(L_k(n)) ) f(L_i(n))

should be asymptotically negligible; intuitively, the point here is that the prime factorisation of {L_i(n)} should not influence the Liouville function of {L_j(n)}, even on the short arithmetic progressions that the divisor sum {\nu} is built out of, and so any monomial {\lambda(L_{i_1}(n)) \dots \lambda(L_{i_r}(n))} occurring in {Q( \lambda(L_1(n)), \dots, \lambda(L_k(n)) )} should exhibit strong cancellation for any of the above sums. If one accepts this principle, then all the expressions (1), (2), (3) should be essentially unchanged when {\nu(n)} is replaced by {\nu(n) w(n)}.

Suppose now for sake of contradiction that one could use sieve-theoretic methods to locate an {n} in the support of some sieve weight {\nu(n)} obeying {P( L_1(n),\dots,L_k(n))}. Then, by reweighting all sieve weights by the additional multiplicative factor of {w(n)}, the same arguments should also be able to locate {n} in the support of {\nu(n) w(n)} for which {P( L_1(n),\dots,L_k(n))} holds. But {w} is only supported on those {n} whose Liouville sign pattern is forbidden, a contradiction. \Box

Claim 1 is sharp in the following sense: if the convex hull of the forbidden sign patterns of {P} do not contain the origin, then by the Hahn-Banach theorem (in the hyperplane separation form), there exist real coefficients {c_1,\dots,c_k} such that

\displaystyle  c_1 \epsilon_1 + \dots + c_k \epsilon_k < -c

for all forbidden sign patterns {(\epsilon_1,\dots,\epsilon_k)} and some {c>0}. On the other hand, from Liouville pseudorandomness one expects that

\displaystyle  \sum_n \nu(n) (c_1 \lambda(L_1(n)) + \dots + c_k \lambda(L_k(n)))

is negligible (as compared against {\sum_n \nu(n)} for any reasonable sieve weight {\nu}. We conclude that for some {n} in the support of {\nu}, that

\displaystyle  c_1 \lambda(L_1(n)) + \dots + c_k \lambda(L_k(n)) > -c \ \ \ \ \ (4)

and hence {(\lambda(L_1(n)),\dots,\lambda(L_k(n)))} is not a forbidden sign pattern. This does not actually imply that {P(L_1(n),\dots,L_k(n))} holds, but it does not prevent {P(L_1(n),\dots,L_k(n))} from holding purely from parity considerations. Thus, we do not expect a parity obstruction of the type in Claim 1 to hold when the convex hull of forbidden sign patterns does not contain the origin.

Example 4 Let {G} be a graph on {k} vertices {\{1,\dots,k\}}, and let {P(l_1,\dots,l_k)} be the property that one can find an edge {\{i,j\}} of {G} with {l_i,l_j} both prime. We claim that this property is subject to the parity problem precisely when {G} is two-colourable. Indeed, if {G} is two-colourable, then we can colour {\{1,\dots,k\}} into two colours (say, red and green) such that all edges in {G} connect a red vertex to a green vertex. If we then consider the two sign patterns in which all the red vertices have one sign and the green vertices have the opposite sign, these are two forbidden sign patterns which contain the origin in the convex hull, and so the parity problem applies. Conversely, suppose that {G} is not two-colourable, then it contains an odd cycle. Any forbidden sign pattern then must contain more {+1}s on this odd cycle than {-1}s (since otherwise two of the {-1}s are adjacent on this cycle by the pigeonhole principle, and this is not forbidden), and so by convexity any tuple in the convex hull of this sign pattern has a positive sum on this odd cycle. Hence the origin is not in the convex hull, and the parity obstruction does not apply. (See also this previous post for a similar obstruction ultimately coming from two-colourability).

Example 5 An example of a parity-obstructed property (supplied by Zeb Brady) that does not come from two-colourability: we let {P( l_{\{1,2\}}, l_{\{1,3\}}, l_{\{1,4\}}, l_{\{2,3\}}, l_{\{2,4\}}, l_{\{3,4\}} )} be the property that {l_{A_1},\dots,l_{A_r}} are prime for some collection {A_1,\dots,A_r} of pair sets that cover {\{1,\dots,4\}}. For instance, this property holds if {l_{\{1,2\}}, l_{\{3,4\}}} are both prime, or if {l_{\{1,2\}}, l_{\{1,3\}}, l_{\{1,4\}}} are all prime, but not if {l_{\{1,2\}}, l_{\{1,3\}}, l_{\{2,3\}}} are the only primes. An example of a forbidden sign pattern is the pattern where {\{1,2\}, \{2,3\}, \{1,3\}} are given the sign {-1}, and the other three pairs are given {+1}. Averaging over permutations of {1,2,3,4} we see that zero lies in the convex hull, and so this example is blocked by parity. However, there is no sign pattern such that it and its negation are both forbidden, which is another formulation of two-colourability.

Of course, the absence of a parity obstruction does not automatically mean that the desired claim is true. For instance, given an admissible {5}-tuple {h_1,\dots,h_5}, parity obstructions do not prevent one from establishing the existence of infinitely many {n} such that at least three of {n+h_1,\dots,n+h_5} are prime, however we are not yet able to actually establish this, even assuming strong sieve-theoretic hypotheses such as the generalised Elliott-Halberstam hypothesis. (However, the argument giving (4) does easily give the far weaker claim that there exist infinitely many {n} such that at least three of {n+h_1,\dots,n+h_5} have a Liouville function of {-1}.)

Remark 1 Another way to get past the parity problem in some cases is to take advantage of linear forms that are constant multiples of each other (which correlates the Liouville functions to each other). For instance, on GEH we can find two {E_3} numbers (products of exactly three primes) that differ by exactly {60}; a direct sieve approach using the linear forms {n,n+60} fails due to the parity obstruction, but instead one can first find {n} such that two of {n,n+4,n+10} are prime, and then among the pairs of linear forms {(15n,15n+60)}, {(6n,6n+60)}, {(10n+40,10n+100)} one can find a pair of {E_3} numbers that differ by exactly {60}. See this paper of Goldston, Graham, Pintz, and Yildirim for more examples of this type.

I thank John Friedlander and Sid Graham for helpful discussions and encouragement.

In the winter quarter (starting January 5) I will be teaching a graduate topics course entitled “An introduction to analytic prime number theory“. As the name suggests, this is a course covering many of the analytic number theory techniques used to study the distribution of the prime numbers {{\mathcal P} = \{2,3,5,7,11,\dots\}}. I will list the topics I intend to cover in this course below the fold. As with my previous courses, I will place lecture notes online on my blog in advance of the physical lectures.

The type of results about primes that one aspires to prove here is well captured by Landau’s classical list of problems:

  1. Even Goldbach conjecture: every even number {N} greater than two is expressible as the sum of two primes.
  2. Twin prime conjecture: there are infinitely many pairs {n,n+2} which are simultaneously prime.
  3. Legendre’s conjecture: for every natural number {N}, there is a prime between {N^2} and {(N+1)^2}.
  4. There are infinitely many primes of the form {n^2+1}.

All four of Landau’s problems remain open, but we have convincing heuristic evidence that they are all true, and in each of the four cases we have some highly non-trivial partial results, some of which will be covered in this course. We also now have some understanding of the barriers we are facing to fully resolving each of these problems, such as the parity problem; this will also be discussed in the course.

One of the main reasons that the prime numbers {{\mathcal P}} are so difficult to deal with rigorously is that they have very little usable algebraic or geometric structure that we know how to exploit; for instance, we do not have any useful prime generating functions. One of course can create non-useful functions of this form, such as the ordered parameterisation {n \mapsto p_n} that maps each natural number {n} to the {n^{th}} prime {p_n}, or one could invoke Matiyasevich’s theorem to produce a polynomial of many variables whose only positive values are prime, but these sorts of functions have no usable structure to exploit (for instance, they give no insight into any of the Landau problems listed above; see also Remark 2 below). The various primality tests in the literature, while useful for practical applications (e.g. cryptography) involving primes, have also proven to be of little utility for these sorts of problems; again, see Remark 2. In fact, in order to make plausible heuristic predictions about the primes, it is best to take almost the opposite point of view to the structured viewpoint, using as a starting point the belief that the primes exhibit strong pseudorandomness properties that are largely incompatible with the presence of rigid algebraic or geometric structure. We will discuss such heuristics later in this course.

It may be in the future that some usable structure to the primes (or related objects) will eventually be located (this is for instance one of the motivations in developing a rigorous theory of the “field with one element“, although this theory is far from being fully realised at present). For now, though, analytic and combinatorial methods have proven to be the most effective way forward, as they can often be used even in the near-complete absence of structure.

In this course, we will not discuss combinatorial approaches (such as the deployment of tools from additive combinatorics) in depth, but instead focus on the analytic methods. The basic principles of this approach can be summarised as follows:

  1. Rather than try to isolate individual primes {p} in {{\mathcal P}}, one works with the set of primes {{\mathcal P}} in aggregate, focusing in particular on asymptotic statistics of this set. For instance, rather than try to find a single pair {n,n+2} of twin primes, one can focus instead on the count {|\{ n \leq x: n,n+2 \in {\mathcal P} \}|} of twin primes up to some threshold {x}. Similarly, one can focus on counts such as {|\{ n \leq N: n, N-n \in {\mathcal P} \}|}, {|\{ p \in {\mathcal P}: N^2 < p < (N+1)^2 \}|}, or {|\{ n \leq x: n^2 + 1 \in {\mathcal P} \}|}, which are the natural counts associated to the other three Landau problems. In all four of Landau’s problems, the basic task is now to obtain a non-trivial lower bounds on these counts.
  2. If one wishes to proceed analytically rather than combinatorially, one should convert all these counts into sums, using the fundamental identity

    \displaystyle |A| = \sum_n 1_A(n),

    (or variants thereof) for the cardinality {|A|} of subsets {A} of the natural numbers {{\bf N}}, where {1_A} is the indicator function of {A} (and {n} ranges over {{\bf N}}). Thus we are now interested in estimating (and particularly in lower bounding) sums such as

    \displaystyle \sum_{n \leq N} 1_{{\mathcal P}}(n) 1_{{\mathcal P}}(N-n),

    \displaystyle \sum_{n \leq x} 1_{{\mathcal P}}(n) 1_{{\mathcal P}}(n+2),

    \displaystyle \sum_{N^2 < n < (N+1)^2} 1_{{\mathcal P}}(n),

    or

    \displaystyle \sum_{n \leq x} 1_{{\mathcal P}}(n^2+1).

  3. Once one expresses number-theoretic problems in this fashion, we are naturally led to the more general question of how to accurately estimate (or, less ambitiously, to lower bound or upper bound) sums such as

    \displaystyle \sum_n f(n)

    or more generally bilinear or multilinear sums such as

    \displaystyle \sum_n \sum_m f(n,m)

    or

    \displaystyle \sum_{n_1,\dots,n_k} f(n_1,\dots,n_k)

    for various functions {f} of arithmetic interest. (Importantly, one should also generalise to include integrals as well as sums, particularly contour integrals or integrals over the unit circle or real line, but we postpone discussion of these generalisations to later in the course.) Indeed, a huge portion of modern analytic number theory is devoted to precisely this sort of question. In many cases, we can predict an expected main term for such sums, and then the task is to control the error term between the true sum and its expected main term. It is often convenient to normalise the expected main term to be zero or negligible (e.g. by subtracting a suitable constant from {f}), so that one is now trying to show that a sum of signed real numbers (or perhaps complex numbers) is small. In other words, the question becomes one of rigorously establishing a significant amount of cancellation in one’s sums (also referred to as a gain or savings over a benchmark “trivial bound”). Or to phrase it negatively, the task is to rigorously prevent a conspiracy of non-cancellation, caused for instance by two factors in the summand {f(n)} exhibiting an unexpectedly large correlation with each other.

  4. It is often difficult to discern cancellation (or to prevent conspiracy) directly for a given sum (such as {\sum_n f(n)}) of interest. However, analytic number theory has developed a large number of techniques to relate one sum to another, and then the strategy is to keep transforming the sum into more and more analytically tractable expressions, until one arrives at a sum for which cancellation can be directly exhibited. (Note though that there is often a short-term tradeoff between analytic tractability and algebraic simplicity; in a typical analytic number theory argument, the sums will get expanded and decomposed into many quite messy-looking sub-sums, until at some point one applies some crude estimation to replace these messy sub-sums by tractable ones again.) There are many transformations available, ranging such basic tools as the triangle inequality, pointwise domination, or the Cauchy-Schwarz inequality to key identities such as multiplicative number theory identities (such as the Vaughan identity and the Heath-Brown identity), Fourier-analytic identities (e.g. Fourier inversion, Poisson summation, or more advanced trace formulae), or complex analytic identities (e.g. the residue theorem, Perron’s formula, or Jensen’s formula). The sheer range of transformations available can be intimidating at first; there is no shortage of transformations and identities in this subject, and if one applies them randomly then one will typically just transform a difficult sum into an even more difficult and intractable expression. However, one can make progress if one is guided by the strategy of isolating and enhancing a desired cancellation (or conspiracy) to the point where it can be easily established (or dispelled), or alternatively to reach the point where no deep cancellation is needed for the application at hand (or equivalently, that no deep conspiracy can disrupt the application).
  5. One particularly powerful technique (albeit one which, ironically, can be highly “ineffective” in a certain technical sense to be discussed later) is to use one potential conspiracy to defeat another, a technique I refer to as the “dueling conspiracies” method. This technique may be unable to prevent a single strong conspiracy, but it can sometimes be used to prevent two or more such conspiracies from occurring, which is particularly useful if conspiracies come in pairs (e.g. through complex conjugation symmetry, or a functional equation). A related (but more “effective”) strategy is to try to “disperse” a single conspiracy into several distinct conspiracies, which can then be used to defeat each other.

As stated before, the above strategy has not been able to establish any of the four Landau problems as stated. However, they can come close to such problems (and we now have some understanding as to why these problems remain out of reach of current methods). For instance, by using these techniques (and a lot of additional effort) one can obtain the following sample partial results in the Landau problems:

  1. Chen’s theorem: every sufficiently large even number {N} is expressible as the sum of a prime and an almost prime (the product of at most two primes). The proof proceeds by finding a nontrivial lower bound on {\sum_{n \leq N} 1_{\mathcal P}(n) 1_{{\mathcal E}_2}(N-n)}, where {{\mathcal E}_2} is the set of almost primes.
  2. Zhang’s theorem: There exist infinitely many pairs {p_n, p_{n+1}} of consecutive primes with {p_{n+1} - p_n \leq 7 \times 10^7}. The proof proceeds by giving a non-negative lower bound on the quantity {\sum_{x \leq n \leq 2x} (\sum_{i=1}^k 1_{\mathcal P}(n+h_i) - 1)} for large {x} and certain distinct integers {h_1,\dots,h_k} between {0} and {7 \times 10^7}. (The bound {7 \times 10^7} has since been lowered to {246}.)
  3. The Baker-Harman-Pintz theorem: for sufficiently large {x}, there is a prime between {x} and {x + x^{0.525}}. Proven by finding a nontrivial lower bound on {\sum_{x \leq n \leq x+x^{0.525}} 1_{\mathcal P}(n)}.
  4. The Friedlander-Iwaniec theorem: There are infinitely many primes of the form {n^2+m^4}. Proven by finding a nontrivial lower bound on {\sum_{n,m: n^2+m^4 \leq x} 1_{{\mathcal P}}(n^2+m^4)}.

We will discuss (simpler versions of) several of these results in this course.

Of course, for the above general strategy to have any chance of succeeding, one must at some point use some information about the set {{\mathcal P}} of primes. As stated previously, usefully structured parametric descriptions of {{\mathcal P}} do not appear to be available. However, we do have two other fundamental and useful ways to describe {{\mathcal P}}:

  1. (Sieve theory description) The primes {{\mathcal P}} consist of those numbers greater than one, that are not divisible by any smaller prime.
  2. (Multiplicative number theory description) The primes {{\mathcal P}} are the multiplicative generators of the natural numbers {{\bf N}}: every natural number is uniquely factorisable (up to permutation) into the product of primes (the fundamental theorem of arithmetic).

The sieve-theoretic description and its variants lead one to a good understanding of the almost primes, which turn out to be excellent tools for controlling the primes themselves, although there are known limitations as to how much information on the primes one can extract from sieve-theoretic methods alone, which we will discuss later in this course. The multiplicative number theory methods lead one (after some complex or Fourier analysis) to the Riemann zeta function (and other L-functions, particularly the Dirichlet L-functions), with the distribution of zeroes (and poles) of these functions playing a particularly decisive role in the multiplicative methods.

Many of our strongest results in analytic prime number theory are ultimately obtained by incorporating some combination of the above two fundamental descriptions of {{\mathcal P}} (or variants thereof) into the general strategy described above. In contrast, more advanced descriptions of {{\mathcal P}}, such as those coming from the various primality tests available, have (until now, at least) been surprisingly ineffective in practice for attacking problems such as Landau’s problems. One reason for this is that such tests generally involve operations such as exponentiation {a \mapsto a^n} or the factorial function {n \mapsto n!}, which grow too quickly to be amenable to the analytic techniques discussed above.

To give a simple illustration of these two basic approaches to the primes, let us first give two variants of the usual proof of Euclid’s theorem:

Theorem 1 (Euclid’s theorem) There are infinitely many primes.

Proof: (Multiplicative number theory proof) Suppose for contradiction that there were only finitely many primes {p_1,\dots,p_n}. Then, by the fundamental theorem of arithmetic, every natural number is expressible as the product of the primes {p_1,\dots,p_n}. But the natural number {p_1 \dots p_n + 1} is larger than one, but not divisible by any of the primes {p_1,\dots,p_n}, a contradiction.

(Sieve-theoretic proof) Suppose for contradiction that there were only finitely many primes {p_1,\dots,p_n}. Then, by the Chinese remainder theorem, the set of natural numbers {A} that is not divisible by any of the {p_1,\dots,p_n} has density {\prod_{i=1}^n (1-\frac{1}{p_i})}, that is to say

\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N} | A \cap \{1,\dots,N\} | = \prod_{i=1}^n (1-\frac{1}{p_i}).

In particular, {A} has positive density and thus contains an element larger than {1}. But the least such element is one further prime in addition to {p_1,\dots,p_n}, a contradiction. \Box

Remark 1 One can also phrase the proof of Euclid’s theorem in a fashion that largely avoids the use of contradiction; see this previous blog post for more discussion.

Both proofs in fact extend to give a stronger result:

Theorem 2 (Euler’s theorem) The sum {\sum_{p \in {\mathcal P}} \frac{1}{p}} is divergent.

Proof: (Multiplicative number theory proof) By the fundamental theorem of arithmetic, every natural number is expressible uniquely as the product {p_1^{a_1} \dots p_n^{a_n}} of primes in increasing order. In particular, we have the identity

\displaystyle \sum_{n=1}^\infty \frac{1}{n} = \prod_{p \in {\mathcal P}} ( 1 + \frac{1}{p} + \frac{1}{p^2} + \dots )

(both sides make sense in {[0,+\infty]} as everything is unsigned). Since the left-hand side is divergent, the right-hand side is as well. But

\displaystyle ( 1 + \frac{1}{p} + \frac{1}{p^2} + \dots ) = \exp( \frac{1}{p} + O( \frac{1}{p^2} ) )

and {\sum_{p \in {\mathcal P}} \frac{1}{p^2}\leq \sum_{n=1}^\infty \frac{1}{n^2} < \infty}, so {\sum_{p \in {\mathcal P}} \frac{1}{p}} must be divergent.

(Sieve-theoretic proof) Suppose for contradiction that the sum {\sum_{p \in {\mathcal P}} \frac{1}{p}} is convergent. For each natural number {k}, let {A_k} be the set of natural numbers not divisible by the first {k} primes {p_1,\dots,p_k}, and let {A} be the set of numbers not divisible by any prime in {{\mathcal P}}. As in the previous proof, each {A_k} has density {\prod_{i=1}^k (1-\frac{1}{p_i})}. Also, since {\{1,\dots,N\}} contains at most {\frac{N}{p}} multiples of {p}, we have from the union bound that

\displaystyle | A \cap \{1,\dots,N \}| = |A_k \cap \{1,\dots,N\}| - O( N \sum_{i > k} \frac{1}{p_i} ).

Since {\sum_{i=1}^\infty \frac{1}{p_i}} is assumed to be convergent, we conclude that the density of {A_k} converges to the density of {A}; thus {A} has density {\prod_{i=1}^\infty (1-\frac{1}{p_i})}, which is non-zero by the hypothesis that {\sum_{i=1}^\infty \frac{1}{p_i}} converges. On the other hand, since the primes are the only numbers greater than one not divisible by smaller primes, {A} is just {\{1\}}, which has density zero, giving the desired contradiction. \Box

Remark 2 We have seen how easy it is to prove Euler’s theorem by analytic methods. In contrast, there does not seem to be any known proof of this theorem that proceeds by using any sort of prime-generating formula or a primality test, which is further evidence that such tools are not the most effective way to make progress on problems such as Landau’s problems. (But the weaker theorem of Euclid, Theorem 1, can sometimes be proven by such devices.)

The two proofs of Theorem 2 given above are essentially the same proof, as is hinted at by the geometric series identity

\displaystyle 1 + \frac{1}{p} + \frac{1}{p^2} + \dots = (1 - \frac{1}{p})^{-1}.

One can also see the Riemann zeta function begin to make an appearance in both proofs. Once one goes beyond Euler’s theorem, though, the sieve-theoretic and multiplicative methods begin to diverge significantly. On one hand, sieve theory can still handle to some extent sets such as twin primes, despite the lack of multiplicative structure (one simply has to sieve out two residue classes per prime, rather than one); on the other, multiplicative number theory can attain results such as the prime number theorem for which purely sieve theoretic techniques have not been able to establish. The deepest results in analytic number theory will typically require a combination of both sieve-theoretic methods and multiplicative methods in conjunction with the many transforms discussed earlier (and, in many cases, additional inputs from other fields of mathematics such as arithmetic geometry, ergodic theory, or additive combinatorics).

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The wave equation is usually expressed in the form

\displaystyle  \partial_{tt} u - \Delta u = 0

where {u \colon {\bf R} \times {\bf R}^d \rightarrow {\bf C}} is a function of both time {t \in {\bf R}} and space {x \in {\bf R}^d}, with {\Delta} being the Laplacian operator. One can generalise this equation in a number of ways, for instance by replacing the spatial domain {{\bf R}^d} with some other manifold and replacing the Laplacian {\Delta} with the Laplace-Beltrami operator or adding lower order terms (such as a potential, or a coupling with a magnetic field). But for sake of discussion let us work with the classical wave equation on {{\bf R}^d}. We will work formally in this post, being unconcerned with issues of convergence, justifying interchange of integrals, derivatives, or limits, etc.. One then has a conserved energy

\displaystyle  \int_{{\bf R}^d} \frac{1}{2} |\nabla u(t,x)|^2 + \frac{1}{2} |\partial_t u(t,x)|^2\ dx

which we can rewrite using integration by parts and the {L^2} inner product {\langle, \rangle} on {{\bf R}^d} as

\displaystyle  \frac{1}{2} \langle -\Delta u(t), u(t) \rangle + \frac{1}{2} \langle \partial_t u(t), \partial_t u(t) \rangle.

A key feature of the wave equation is finite speed of propagation: if, at time {t=0} (say), the initial position {u(0)} and initial velocity {\partial_t u(0)} are both supported in a ball {B(x_0,R) := \{ x \in {\bf R}^d: |x-x_0| \leq R \}}, then at any later time {t>0}, the position {u(t)} and velocity {\partial_t u(t)} are supported in the larger ball {B(x_0,R+t)}. This can be seen for instance (formally, at least) by inspecting the exterior energy

\displaystyle  \int_{|x-x_0| > R+t} \frac{1}{2} |\nabla u(t,x)|^2 + \frac{1}{2} |\partial_t u(t,x)|^2\ dx

and observing (after some integration by parts and differentiation under the integral sign) that it is non-increasing in time, non-negative, and vanishing at time {t=0}.

The wave equation is second order in time, but one can turn it into a first order system by working with the pair {(u(t),v(t))} rather than just the single field {u(t)}, where {v(t) := \partial_t u(t)} is the velocity field. The system is then

\displaystyle  \partial_t u(t) = v(t)

\displaystyle  \partial_t v(t) = \Delta u(t)

and the conserved energy is now

\displaystyle  \frac{1}{2} \langle -\Delta u(t), u(t) \rangle + \frac{1}{2} \langle v(t), v(t) \rangle. \ \ \ \ \ (1)

Finite speed of propagation then tells us that if {u(0),v(0)} are both supported on {B(x_0,R)}, then {u(t),v(t)} are supported on {B(x_0,R+t)} for all {t>0}. One also has time reversal symmetry: if {t \mapsto (u(t),v(t))} is a solution, then {t \mapsto (u(-t), -v(-t))} is a solution also, thus for instance one can establish an analogue of finite speed of propagation for negative times {t<0} using this symmetry.

If one has an eigenfunction

\displaystyle  -\Delta \phi = \lambda^2 \phi

of the Laplacian, then we have the explicit solutions

\displaystyle  u(t) = e^{\pm it \lambda} \phi

\displaystyle  v(t) = \pm i \lambda e^{\pm it \lambda} \phi

of the wave equation, which formally can be used to construct all other solutions via the principle of superposition.

When one has vanishing initial velocity {v(0)=0}, the solution {u(t)} is given via functional calculus by

\displaystyle  u(t) = \cos(t \sqrt{-\Delta}) u(0)

and the propagator {\cos(t \sqrt{-\Delta})} can be expressed as the average of half-wave operators:

\displaystyle  \cos(t \sqrt{-\Delta}) = \frac{1}{2} ( e^{it\sqrt{-\Delta}} + e^{-it\sqrt{-\Delta}} ).

One can view {\cos(t \sqrt{-\Delta} )} as a minor of the full wave propagator

\displaystyle  U(t) := \exp \begin{pmatrix} 0 & t \\ t\Delta & 0 \end{pmatrix}

\displaystyle  = \begin{pmatrix} \cos(t \sqrt{-\Delta}) & \frac{\sin(t\sqrt{-\Delta})}{\sqrt{-\Delta}} \\ \sin(t\sqrt{-\Delta}) \sqrt{-\Delta} & \cos(t \sqrt{-\Delta} ) \end{pmatrix}

which is unitary with respect to the energy form (1), and is the fundamental solution to the wave equation in the sense that

\displaystyle  \begin{pmatrix} u(t) \\ v(t) \end{pmatrix} = U(t) \begin{pmatrix} u(0) \\ v(0) \end{pmatrix}. \ \ \ \ \ (2)

Viewing the contraction {\cos(t\sqrt{-\Delta})} as a minor of a unitary operator is an instance of the “dilation trick“.

It turns out (as I learned from Yuval Peres) that there is a useful discrete analogue of the wave equation (and of all of the above facts), in which the time variable {t} now lives on the integers {{\bf Z}} rather than on {{\bf R}}, and the spatial domain can be replaced by discrete domains also (such as graphs). Formally, the system is now of the form

\displaystyle  u(t+1) = P u(t) + v(t) \ \ \ \ \ (3)

\displaystyle  v(t+1) = P v(t) - (1-P^2) u(t)

where {t} is now an integer, {u(t), v(t)} take values in some Hilbert space (e.g. {\ell^2} functions on a graph {G}), and {P} is some operator on that Hilbert space (which in applications will usually be a self-adjoint contraction). To connect this with the classical wave equation, let us first consider a rescaling of this system

\displaystyle  u(t+\varepsilon) = P_\varepsilon u(t) + \varepsilon v(t)

\displaystyle  v(t+\varepsilon) = P_\varepsilon v(t) - \frac{1}{\varepsilon} (1-P_\varepsilon^2) u(t)

where {\varepsilon>0} is a small parameter (representing the discretised time step), {t} now takes values in the integer multiples {\varepsilon {\bf Z}} of {\varepsilon}, and {P_\varepsilon} is the wave propagator operator {P_\varepsilon := \cos( \varepsilon \sqrt{-\Delta} )} or the heat propagator {P_\varepsilon := \exp( - \varepsilon^2 \Delta/2 )} (the two operators are different, but agree to fourth order in {\varepsilon}). One can then formally verify that the wave equation emerges from this rescaled system in the limit {\varepsilon \rightarrow 0}. (Thus, {P} is not exactly the direct analogue of the Laplacian {\Delta}, but can be viewed as something like {P_\varepsilon = 1 - \frac{\varepsilon^2}{2} \Delta + O( \varepsilon^4 )} in the case of small {\varepsilon}, or {P = 1 - \frac{1}{2}\Delta + O(\Delta^2)} if we are not rescaling to the small {\varepsilon} case. The operator {P} is sometimes known as the diffusion operator)

Assuming {P} is self-adjoint, solutions to the system (3) formally conserve the energy

\displaystyle  \frac{1}{2} \langle (1-P^2) u(t), u(t) \rangle + \frac{1}{2} \langle v(t), v(t) \rangle. \ \ \ \ \ (4)

This energy is positive semi-definite if {P} is a contraction. We have the same time reversal symmetry as before: if {t \mapsto (u(t),v(t))} solves the system (3), then so does {t \mapsto (u(-t), -v(-t))}. If one has an eigenfunction

\displaystyle  P \phi = \cos(\lambda) \phi

to the operator {P}, then one has an explicit solution

\displaystyle  u(t) = e^{\pm it \lambda} \phi

\displaystyle  v(t) = \pm i \sin(\lambda) e^{\pm it \lambda} \phi

to (3), and (in principle at least) this generates all other solutions via the principle of superposition.

Finite speed of propagation is a lot easier in the discrete setting, though one has to offset the support of the “velocity” field {v} by one unit. Suppose we know that {P} has unit speed in the sense that whenever {f} is supported in a ball {B(x,R)}, then {Pf} is supported in the ball {B(x,R+1)}. Then an easy induction shows that if {u(0), v(0)} are supported in {B(x_0,R), B(x_0,R+1)} respectively, then {u(t), v(t)} are supported in {B(x_0,R+t), B(x_0, R+t+1)}.

The fundamental solution {U(t) = U^t} to the discretised wave equation (3), in the sense of (2), is given by the formula

\displaystyle  U(t) = U^t = \begin{pmatrix} P & 1 \\ P^2-1 & P \end{pmatrix}^t

\displaystyle  = \begin{pmatrix} T_t(P) & U_{t-1}(P) \\ (P^2-1) U_{t-1}(P) & T_t(P) \end{pmatrix}

where {T_t} and {U_t} are the Chebyshev polynomials of the first and second kind, thus

\displaystyle  T_t( \cos \theta ) = \cos(t\theta)

and

\displaystyle  U_t( \cos \theta ) = \frac{\sin((t+1)\theta)}{\sin \theta}.

In particular, {P} is now a minor of {U(1) = U}, and can also be viewed as an average of {U} with its inverse {U^{-1}}:

\displaystyle  \begin{pmatrix} P & 0 \\ 0 & P \end{pmatrix} = \frac{1}{2} (U + U^{-1}). \ \ \ \ \ (5)

As before, {U} is unitary with respect to the energy form (4), so this is another instance of the dilation trick in action. The powers {P^n} and {U^n} are discrete analogues of the heat propagators {e^{t\Delta/2}} and wave propagators {U(t)} respectively.

One nice application of all this formalism, which I learned from Yuval Peres, is the Varopoulos-Carne inequality:

Theorem 1 (Varopoulos-Carne inequality) Let {G} be a (possibly infinite) regular graph, let {n \geq 1}, and let {x, y} be vertices in {G}. Then the probability that the simple random walk at {x} lands at {y} at time {n} is at most {2 \exp( - d(x,y)^2 / 2n )}, where {d} is the graph distance.

This general inequality is quite sharp, as one can see using the standard Cayley graph on the integers {{\bf Z}}. Very roughly speaking, it asserts that on a regular graph of reasonably controlled growth (e.g. polynomial growth), random walks of length {n} concentrate on the ball of radius {O(\sqrt{n})} or so centred at the origin of the random walk.

Proof: Let {P \colon \ell^2(G) \rightarrow \ell^2(G)} be the graph Laplacian, thus

\displaystyle  Pf(x) = \frac{1}{D} \sum_{y \sim x} f(y)

for any {f \in \ell^2(G)}, where {D} is the degree of the regular graph and sum is over the {D} vertices {y} that are adjacent to {x}. This is a contraction of unit speed, and the probability that the random walk at {x} lands at {y} at time {n} is

\displaystyle  \langle P^n \delta_x, \delta_y \rangle

where {\delta_x, \delta_y} are the Dirac deltas at {x,y}. Using (5), we can rewrite this as

\displaystyle  \langle (\frac{1}{2} (U + U^{-1}))^n \begin{pmatrix} 0 \\ \delta_x\end{pmatrix}, \begin{pmatrix} 0 \\ \delta_y\end{pmatrix} \rangle

where we are now using the energy form (4). We can write

\displaystyle  (\frac{1}{2} (U + U^{-1}))^n = {\bf E} U^{S_n}

where {S_n} is the simple random walk of length {n} on the integers, that is to say {S_n = \xi_1 + \dots + \xi_n} where {\xi_1,\dots,\xi_n = \pm 1} are independent uniform Bernoulli signs. Thus we wish to show that

\displaystyle  {\bf E} \langle U^{S_n} \begin{pmatrix} 0 \\ \delta_x\end{pmatrix}, \begin{pmatrix} 0 \\ \delta_y\end{pmatrix} \rangle \leq 2 \exp(-d(x,y)^2 / 2n ).

By finite speed of propagation, the inner product here vanishes if {|S_n| < d(x,y)}. For {|S_n| \geq d(x,y)} we can use Cauchy-Schwarz and the unitary nature of {U} to bound the inner product by {1}. Thus the left-hand side may be upper bounded by

\displaystyle  {\bf P}( |S_n| \geq d(x,y) )

and the claim now follows from the Chernoff inequality. \Box

This inequality has many applications, particularly with regards to relating the entropy, mixing time, and concentration of random walks with volume growth of balls; see this text of Lyons and Peres for some examples.

For sake of comparison, here is a continuous counterpart to the Varopoulos-Carne inequality:

Theorem 2 (Continuous Varopoulos-Carne inequality) Let {t > 0}, and let {f,g \in L^2({\bf R}^d)} be supported on compact sets {F,G} respectively. Then

\displaystyle  |\langle e^{t\Delta/2} f, g \rangle| \leq \sqrt{\frac{2t}{\pi d(F,G)^2}} \exp( - d(F,G)^2 / 2t ) \|f\|_{L^2} \|g\|_{L^2}

where {d(F,G)} is the Euclidean distance between {F} and {G}.

Proof: By Fourier inversion one has

\displaystyle  e^{-t\xi^2/2} = \frac{1}{\sqrt{2\pi t}} \int_{\bf R} e^{-s^2/2t} e^{is\xi}\ ds

\displaystyle  = \sqrt{\frac{2}{\pi t}} \int_0^\infty e^{-s^2/2t} \cos(s \xi )\ ds

for any real {\xi}, and thus

\displaystyle  \langle e^{t\Delta/2} f, g\rangle = \sqrt{\frac{2}{\pi}} \int_0^\infty e^{-s^2/2t} \langle \cos(s \sqrt{-\Delta} ) f, g \rangle\ ds.

By finite speed of propagation, the inner product {\langle \cos(s \sqrt{-\Delta} ) f, g \rangle\ ds} vanishes when {s < d(F,G)}; otherwise, we can use Cauchy-Schwarz and the contractive nature of {\cos(s \sqrt{-\Delta} )} to bound this inner product by {\|f\|_{L^2} \|g\|_{L^2}}. Thus

\displaystyle  |\langle e^{t\Delta/2} f, g\rangle| \leq \sqrt{\frac{2}{\pi t}} \|f\|_{L^2} \|g\|_{L^2} \int_{d(F,G)}^\infty e^{-s^2/2t}\ ds.

Bounding {e^{-s^2/2t}} by {e^{-d(F,G)^2/2t} e^{-d(F,G) (s-d(F,G))/t}}, we obtain the claim. \Box

Observe that the argument is quite general and can be applied for instance to other Riemannian manifolds than {{\bf R}^d}.

I’ve just uploaded to the arXiv my paper “The Elliott-Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture“. As the title suggests, this paper links together the Elliott-Halberstam conjecture from sieve theory with the conjecture of Vinogradov concerning the least quadratic nonresidue {n(p)} of a prime {p}. Vinogradov established the bound

\displaystyle  n(p) \ll p^{\frac{1}{2\sqrt{e}}} \log^2 p \ \ \ \ \ (1)

and conjectured that

\displaystyle  n(p) \ll p^\varepsilon \ \ \ \ \ (2)

for any fixed {\varepsilon>0}. Unconditionally, the best result so far (up to logarithmic factors) that holds for all primes {p} is by Burgess, who showed that

\displaystyle  n(p) \ll p^{\frac{1}{4\sqrt{e}}+\varepsilon} \ \ \ \ \ (3)

for any fixed {\varepsilon>0}. See this previous post for a proof of these bounds.

In this paper, we show that the Vinogradov conjecture is a consequence of the Elliott-Halberstam conjecture. Using a variant of the argument, we also show that the “Type II” estimates established by Zhang and numerically improved by the Polymath8a project can be used to improve a little on the Vinogradov bound (1), to

\displaystyle  n(p) \ll p^{(\frac{1}{2}-\frac{1}{34})\frac{1}{\sqrt{e}} + \varepsilon},

although this falls well short of the Burgess bound. However, the method is somewhat different (although in both cases it is the Weil exponential sum bounds that are the source of the gain over (1)) and it is conceivable that a numerically stronger version of the Type II estimates could obtain results that are more competitive with the Burgess bound. At any rate, this demonstrates that the equidistribution estimates introduced by Zhang may have further applications beyond the family of results related to bounded gaps between primes.

The connection between the least quadratic nonresidue problem and Elliott-Halberstam is follows. Suppose for contradiction we can find a prime {q} with {n(q)} unusually large. Letting {\chi} be the quadratic character modulo {q}, this implies that the sums {\sum_{n \leq x} \chi(n)} are also unusually large for a significant range of {x} (e.g. {x < n(q)}), although the sum is also quite small for large {x} (e.g. {x > q}), due to the cancellation present in {\chi}. It turns out (by a sort of “uncertainty principle” for multiplicative functions, as per this paper of Granville and Soundararajan) that these facts force {\sum_{n\leq x} \chi(n) \Lambda(n)} to be unusually large in magnitude for some large {x} (with {q^C \leq x \leq q^{C'}} for two large absolute constants {C,C'}). By the periodicity of {\chi}, this means that

\displaystyle  \sum_{n\leq x} \chi(n) \Lambda(n+q)

must be unusually large also. However, because {n(q)} is large, one can factorise {\chi} as {f * 1} for a fairly sparsely supported function {f = \chi * \mu}. The Elliott-Halberstam conjecture, which controls the distribution of {\Lambda} in arithmetic progressions on the average can then be used to show that {\sum_{n \leq x} (f*1)(n) \Lambda(n+q)} is small, giving the required contradiction.

The implication involving Type II estimates is proven by a variant of the argument. If {n(q)} is large, then a character sum {\sum_{N\leq n \leq 2N} \chi(n)} is unusually large for a certain {N}. By multiplicativity of {\chi}, this shows that {\chi} correlates with {\chi * 1_{[N,2N]}}, and then by periodicity of {\chi}, this shows that {\chi(n)} correlates with {\chi * 1_{[N,2N]}(n+jq)} for various small {q}. By the Cauchy-Schwarz inequality (cf. this previous blog post), this implies that {\chi * 1_{[N,2N]}(n+jq)} correlates with {\chi * 1_{[N,2N]}(n+j'q)} for some distinct {j,j'}. But this can be ruled out by using Type II estimates.

I’ll record here a well-known observation concerning potential counterexamples to any improvement to the Burgess bound, that is to say an infinite sequence of primes {p} with {n(p) = p^{\frac{1}{4\sqrt{e}} + o(1)}}. Suppose we let {a(t)} be the asymptotic mean value of the quadratic character {\chi} at {p^t} and {b(t)} the mean value of {\chi \Lambda}; these quantities are defined precisely in my paper, but roughly speaking one can think of

\displaystyle  a(t) = \lim_{p \rightarrow \infty} \frac{1}{p^t} \sum_{n \leq p^t} \chi(n)

and

\displaystyle  b(t) = \lim_{p \rightarrow \infty} \frac{1}{p^t} \sum_{n \leq p^t} \chi(n) \Lambda(n).

Thanks to the basic Dirichlet convolution identity {\chi(n) \log(n) = \chi * \chi\Lambda(n)}, one can establish the Wirsing integral equation

\displaystyle  t a(t) = \int_0^t a(u) b(t-u)\ du

for all {t \geq 0}; see my paper for details (actually far sharper claims than this appear in previous work of Wirsing and Granville-Soundararajan). If we have an infinite sequence of counterexamples to any improvement to the Burgess bound, then we have

\displaystyle  a(t)=b(t) = 1 \hbox{ for } t < \frac{1}{4\sqrt{e}}

while from the Burgess exponential sum estimates we have

\displaystyle  a(t) = 0 \hbox{ for } t > \frac{1}{4}.

These two constraints, together with the Wirsing integral equation, end up determining {a} and {b} completely. It turns out that we must have

\displaystyle  b(t) = -1 \hbox{ for } \frac{1}{4\sqrt{e}} \leq t \leq \frac{1}{4}

and

\displaystyle  a(t) = 1 - 2 \log(4 \sqrt{e} t) \hbox{ for } \frac{1}{4\sqrt{e}} \leq t \leq \frac{1}{4}

and then for {t > \frac{1}{4}}, {b} evolves by the integral equation

\displaystyle  b(t) = \int_{1/4\sqrt{e}}^{1/4} b(t-u) \frac{2du}{u}.

For instance

\displaystyle  b(t) = 1 \hbox{ for } \frac{1}{4} < t \leq \frac{1}{2\sqrt{e}}

and then {b} oscillates in a somewhat strange fashion towards zero as {t \rightarrow \infty}. This very odd behaviour of {\sum_n \chi(n) \Lambda(n)} is surely impossible, but it seems remarkably hard to exclude it without invoking a strong hypothesis, such as GRH or the Elliott-Halberstam conjecture (or weaker versions thereof).

The prime number theorem can be expressed as the assertion

\displaystyle  \sum_{n \leq x} \Lambda(n) = x + o(x) \ \ \ \ \ (1)

as {x \rightarrow \infty}, where

\displaystyle  \Lambda(n) := \sum_{d|n} \mu(d) \log \frac{n}{d}

is the von Mangoldt function. It is a basic result in analytic number theory, but requires a bit of effort to prove. One “elementary” proof of this theorem proceeds through the Selberg symmetry formula

\displaystyle  \sum_{n \leq x} \Lambda_2(n) = 2 x \log x + O(x) \ \ \ \ \ (2)

where the second von Mangoldt function {\Lambda_2} is defined by the formula

\displaystyle  \Lambda_2(n) := \sum_{d|n} \mu(d) \log^2 \frac{n}{d} \ \ \ \ \ (3)

or equivalently

\displaystyle  \Lambda_2(n) = \Lambda(n) \log n + \sum_{d|n} \Lambda(d) \Lambda(\frac{n}{d}). \ \ \ \ \ (4)

(We are avoiding the use of the {*} symbol here to denote Dirichlet convolution, as we will need this symbol to denote ordinary convolution shortly.) For the convenience of the reader, we give a proof of the Selberg symmetry formula below the fold. Actually, for the purposes of proving the prime number theorem, the weaker estimate

\displaystyle  \sum_{n \leq x} \Lambda_2(n) = 2 x \log x + o(x \log x) \ \ \ \ \ (5)

suffices.

In this post I would like to record a somewhat “soft analysis” reformulation of the elementary proof of the prime number theorem in terms of Banach algebras, and specifically in Banach algebra structures on (completions of) the space {C_c({\bf R})} of compactly supported continuous functions {f: {\bf R} \rightarrow {\bf C}} equipped with the convolution operation

\displaystyle  f*g(t) := \int_{\bf R} f(u) g(t-u)\ du.

This soft argument does not easily give any quantitative decay rate in the prime number theorem, but by the same token it avoids many of the quantitative calculations in the traditional proofs of this theorem. Ultimately, the key “soft analysis” fact used is the spectral radius formula

\displaystyle  \lim_{n \rightarrow \infty} \|f^n\|^{1/n} = \sup_{\lambda \in \hat B} |\lambda(f)| \ \ \ \ \ (6)

for any element {f} of a unital commutative Banach algebra {B}, where {\hat B} is the space of characters (i.e., continuous unital algebra homomorphisms from {B} to {{\bf C}}) of {B}. This formula is due to Gelfand and may be found in any text on Banach algebras; for sake of completeness we prove it below the fold.

The connection between prime numbers and Banach algebras is given by the following consequence of the Selberg symmetry formula.

Theorem 1 (Construction of a Banach algebra norm) For any {G \in C_c({\bf R})}, let {\|G\|} denote the quantity

\displaystyle  \|G\| := \limsup_{x \rightarrow \infty} |\sum_n \frac{\Lambda(n)}{n} G( \log \frac{x}{n} ) - \int_{\bf R} G(t)\ dt|.

Then {\| \|} is a seminorm on {C_c({\bf R})} with the bound

\displaystyle  \|G\| \leq \|G\|_{L^1({\bf R})} := \int_{\bf R} |G(t)|\ dt \ \ \ \ \ (7)

for all {G \in C_c({\bf R})}. Furthermore, we have the Banach algebra bound

\displaystyle  \| G * H \| \leq \|G\| \|H\| \ \ \ \ \ (8)

for all {G,H \in C_c({\bf R})}.

We prove this theorem below the fold. The prime number theorem then follows from Theorem 1 and the following two assertions. The first is an application of the spectral radius formula (6) and some basic Fourier analysis (in particular, the observation that {C_c({\bf R})} contains a plentiful supply of local units:

Theorem 2 (Non-trivial Banach algebras with many local units have non-trivial spectrum) Let {\| \|} be a seminorm on {C_c({\bf R})} obeying (7), (8). Suppose that {\| \|} is not identically zero. Then there exists {\xi \in {\bf R}} such that

\displaystyle  |\int_{\bf R} G(t) e^{-it\xi}\ dt| \leq \|G\|

for all {G \in C_c}. In particular, by (7), one has

\displaystyle  \|G\| = \| G \|_{L^1({\bf R})}

whenever {G(t) e^{-it\xi}} is a non-negative function.

The second is a consequence of the Selberg symmetry formula and the fact that {\Lambda} is real (as well as Mertens’ theorem, in the {\xi=0} case), and is closely related to the non-vanishing of the Riemann zeta function {\zeta} on the line {\{ 1+i\xi: \xi \in {\bf R}\}}:

Theorem 3 (Breaking the parity barrier) Let {\xi \in {\bf R}}. Then there exists {G \in C_c({\bf R})} such that {G(t) e^{-it\xi}} is non-negative, and

\displaystyle  \|G\| < \|G\|_{L^1({\bf R})}.

Assuming Theorems 1, 2, 3, we may now quickly establish the prime number theorem as follows. Theorem 2 and Theorem 3 imply that the seminorm {\| \|} constructed in Theorem 1 is trivial, and thus

\displaystyle  \sum_n \frac{\Lambda(n)}{n} G( \log \frac{x}{n} ) = \int_{\bf R} G(t)\ dt + o(1)

as {x \rightarrow \infty} for any Schwartz function {G} (the decay rate in {o(1)} may depend on {G}). Specialising to functions of the form {G(t) = e^{-t} \eta( e^{-t} )} for some smooth compactly supported {\eta} on {(0,+\infty)}, we conclude that

\displaystyle  \sum_n \Lambda(n) \eta(\frac{n}{x}) = \int_{\bf R} \eta(u)\ du + o(x)

as {x \rightarrow \infty}; by the smooth Urysohn lemma this implies that

\displaystyle  \sum_{\varepsilon x \leq n \leq x} \Lambda(n) = x - \varepsilon x + o(x)

as {x \rightarrow \infty} for any fixed {\varepsilon>0}, and the prime number theorem then follows by a telescoping series argument.

The same argument also yields the prime number theorem in arithmetic progressions, or equivalently that

\displaystyle  \sum_{n \leq x} \Lambda(n) \chi(n) = o(x)

for any fixed Dirichlet character {\chi}; the one difference is that the use of Mertens’ theorem is replaced by the basic fact that the quantity {L(1,\chi) = \sum_n \frac{\chi(n)}{n}} is non-vanishing.

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In graph theory, the recently developed theory of graph limits has proven to be a useful tool for analysing large dense graphs, being a convenient reformulation of the Szemerédi regularity lemma. Roughly speaking, the theory asserts that given any sequence {G_n = (V_n, E_n)} of finite graphs, one can extract a subsequence {G_{n_j} = (V_{n_j}, E_{n_j})} which converges (in a specific sense) to a continuous object known as a “graphon” – a symmetric measurable function {p\colon [0,1] \times [0,1] \rightarrow [0,1]}. What “converges” means in this context is that subgraph densities converge to the associated integrals of the graphon {p}. For instance, the edge density

\displaystyle  \frac{1}{|V_{n_j}|^2} |E_{n_j}|

converge to the integral

\displaystyle  \int_0^1 \int_0^1 p(x,y)\ dx dy,

the triangle density

\displaystyle  \frac{1}{|V_{n_j}|^3} \lvert \{ (v_1,v_2,v_3) \in V_{n_j}^3: \{v_1,v_2\}, \{v_2,v_3\}, \{v_3,v_1\} \in E_{n_j} \} \rvert

converges to the integral

\displaystyle  \int_0^1 \int_0^1 \int_0^1 p(x_1,x_2) p(x_2,x_3) p(x_3,x_1)\ dx_1 dx_2 dx_3,

the four-cycle density

\displaystyle  \frac{1}{|V_{n_j}|^4} \lvert \{ (v_1,v_2,v_3,v_4) \in V_{n_j}^4: \{v_1,v_2\}, \{v_2,v_3\}, \{v_3,v_4\}, \{v_4,v_1\} \in E_{n_j} \} \rvert

converges to the integral

\displaystyle  \int_0^1 \int_0^1 \int_0^1 \int_0^1 p(x_1,x_2) p(x_2,x_3) p(x_3,x_4) p(x_4,x_1)\ dx_1 dx_2 dx_3 dx_4,

and so forth. One can use graph limits to prove many results in graph theory that were traditionally proven using the regularity lemma, such as the triangle removal lemma, and can also reduce many asymptotic graph theory problems to continuous problems involving multilinear integrals (although the latter problems are not necessarily easy to solve!). See this text of Lovasz for a detailed study of graph limits and their applications.

One can also express graph limits (and more generally hypergraph limits) in the language of nonstandard analysis (or of ultraproducts); see for instance this paper of Elek and Szegedy, Section 6 of this previous blog post, or this paper of Towsner. (In this post we assume some familiarity with nonstandard analysis, as reviewed for instance in the previous blog post.) Here, one starts as before with a sequence {G_n = (V_n,E_n)} of finite graphs, and then takes an ultraproduct (with respect to some arbitrarily chosen non-principal ultrafilter {\alpha \in\beta {\bf N} \backslash {\bf N}}) to obtain a nonstandard graph {G_\alpha = (V_\alpha,E_\alpha)}, where {V_\alpha = \prod_{n\rightarrow \alpha} V_n} is the ultraproduct of the {V_n}, and similarly for the {E_\alpha}. The set {E_\alpha} can then be viewed as a symmetric subset of {V_\alpha \times V_\alpha} which is measurable with respect to the Loeb {\sigma}-algebra {{\mathcal L}_{V_\alpha \times V_\alpha}} of the product {V_\alpha \times V_\alpha} (see this previous blog post for the construction of Loeb measure). A crucial point is that this {\sigma}-algebra is larger than the product {{\mathcal L}_{V_\alpha} \times {\mathcal L}_{V_\alpha}} of the Loeb {\sigma}-algebra of the individual vertex set {V_\alpha}. This leads to a decomposition

\displaystyle  1_{E_\alpha} = p + e

where the “graphon” {p} is the orthogonal projection of {1_{E_\alpha}} onto {L^2( {\mathcal L}_{V_\alpha} \times {\mathcal L}_{V_\alpha} )}, and the “regular error” {e} is orthogonal to all product sets {A \times B} for {A, B \in {\mathcal L}_{V_\alpha}}. The graphon {p\colon V_\alpha \times V_\alpha \rightarrow [0,1]} then captures the statistics of the nonstandard graph {G_\alpha}, in exact analogy with the more traditional graph limits: for instance, the edge density

\displaystyle  \hbox{st} \frac{1}{|V_\alpha|^2} |E_\alpha|

(or equivalently, the limit of the {\frac{1}{|V_n|^2} |E_n|} along the ultrafilter {\alpha}) is equal to the integral

\displaystyle  \int_{V_\alpha} \int_{V_\alpha} p(x,y)\ d\mu_{V_\alpha}(x) d\mu_{V_\alpha}(y)

where {d\mu_V} denotes Loeb measure on a nonstandard finite set {V}; the triangle density

\displaystyle  \hbox{st} \frac{1}{|V_\alpha|^3} \lvert \{ (v_1,v_2,v_3) \in V_\alpha^3: \{v_1,v_2\}, \{v_2,v_3\}, \{v_3,v_1\} \in E_\alpha \} \rvert

(or equivalently, the limit along {\alpha} of the triangle densities of {E_n}) is equal to the integral

\displaystyle  \int_{V_\alpha} \int_{V_\alpha} \int_{V_\alpha} p(x_1,x_2) p(x_2,x_3) p(x_3,x_1)\ d\mu_{V_\alpha}(x_1) d\mu_{V_\alpha}(x_2) d\mu_{V_\alpha}(x_3),

and so forth. Note that with this construction, the graphon {p} is living on the Cartesian square of an abstract probability space {V_\alpha}, which is likely to be inseparable; but it is possible to cut down the Loeb {\sigma}-algebra on {V_\alpha} to minimal countable {\sigma}-algebra for which {p} remains measurable (up to null sets), and then one can identify {V_\alpha} with {[0,1]}, bringing this construction of a graphon in line with the traditional notion of a graphon. (See Remark 5 of this previous blog post for more discussion of this point.)

Additive combinatorics, which studies things like the additive structure of finite subsets {A} of an abelian group {G = (G,+)}, has many analogies and connections with asymptotic graph theory; in particular, there is the arithmetic regularity lemma of Green which is analogous to the graph regularity lemma of Szemerédi. (There is also a higher order arithmetic regularity lemma analogous to hypergraph regularity lemmas, but this is not the focus of the discussion here.) Given this, it is natural to suspect that there is a theory of “additive limits” for large additive sets of bounded doubling, analogous to the theory of graph limits for large dense graphs. The purpose of this post is to record a candidate for such an additive limit. This limit can be used as a substitute for the arithmetic regularity lemma in certain results in additive combinatorics, at least if one is willing to settle for qualitative results rather than quantitative ones; I give a few examples of this below the fold.

It seems that to allow for the most flexible and powerful manifestation of this theory, it is convenient to use the nonstandard formulation (among other things, it allows for full use of the transfer principle, whereas a more traditional limit formulation would only allow for a transfer of those quantities continuous with respect to the notion of convergence). Here, the analogue of a nonstandard graph is an ultra approximate group {A_\alpha} in a nonstandard group {G_\alpha = \prod_{n \rightarrow \alpha} G_n}, defined as the ultraproduct of finite {K}-approximate groups {A_n \subset G_n} for some standard {K}. (A {K}-approximate group {A_n} is a symmetric set containing the origin such that {A_n+A_n} can be covered by {K} or fewer translates of {A_n}.) We then let {O(A_\alpha)} be the external subgroup of {G_\alpha} generated by {A_\alpha}; equivalently, {A_\alpha} is the union of {A_\alpha^m} over all standard {m}. This space has a Loeb measure {\mu_{O(A_\alpha)}}, defined by setting

\displaystyle \mu_{O(A_\alpha)}(E_\alpha) := \hbox{st} \frac{|E_\alpha|}{|A_\alpha|}

whenever {E_\alpha} is an internal subset of {A_\alpha^m} for any standard {m}, and extended to a countably additive measure; the arguments in Section 6 of this previous blog post can be easily modified to give a construction of this measure.

The Loeb measure {\mu_{O(A_\alpha)}} is a translation invariant measure on {O(A_{\alpha})}, normalised so that {A_\alpha} has Loeb measure one. As such, one should think of {O(A_\alpha)} as being analogous to a locally compact abelian group equipped with a Haar measure. It should be noted though that {O(A_\alpha)} is not actually a locally compact group with Haar measure, for two reasons:

  • There is not an obvious topology on {O(A_\alpha)} that makes it simultaneously locally compact, Hausdorff, and {\sigma}-compact. (One can get one or two out of three without difficulty, though.)
  • The addition operation {+\colon O(A_\alpha) \times O(A_\alpha) \rightarrow O(A_\alpha)} is not measurable from the product Loeb algebra {{\mathcal L}_{O(A_\alpha)} \times {\mathcal L}_{O(A_\alpha)}} to {{\mathcal L}_{O(\alpha)}}. Instead, it is measurable from the coarser Loeb algebra {{\mathcal L}_{O(A_\alpha) \times O(A_\alpha)}} to {{\mathcal L}_{O(\alpha)}} (compare with the analogous situation for nonstandard graphs).

Nevertheless, the analogy is a useful guide for the arguments that follow.

Let {L(O(A_\alpha))} denote the space of bounded Loeb measurable functions {f\colon O(A_\alpha) \rightarrow {\bf C}} (modulo almost everywhere equivalence) that are supported on {A_\alpha^m} for some standard {m}; this is a complex algebra with respect to pointwise multiplication. There is also a convolution operation {\star\colon L(O(A_\alpha)) \times L(O(A_\alpha)) \rightarrow L(O(A_\alpha))}, defined by setting

\displaystyle  \hbox{st} f \star \hbox{st} g(x) := \hbox{st} \frac{1}{|A_\alpha|} \sum_{y \in A_\alpha^m} f(y) g(x-y)

whenever {f\colon A_\alpha^m \rightarrow {}^* {\bf C}}, {g\colon A_\alpha^l \rightarrow {}^* {\bf C}} are bounded nonstandard functions (extended by zero to all of {O(A_\alpha)}), and then extending to arbitrary elements of {L(O(A_\alpha))} by density. Equivalently, {f \star g} is the pushforward of the {{\mathcal L}_{O(A_\alpha) \times O(A_\alpha)}}-measurable function {(x,y) \mapsto f(x) g(y)} under the map {(x,y) \mapsto x+y}.

The basic structural theorem is then as follows.

Theorem 1 (Kronecker factor) Let {A_\alpha} be an ultra approximate group. Then there exists a (standard) locally compact abelian group {G} of the form

\displaystyle  G = {\bf R}^d \times {\bf Z}^m \times T

for some standard {d,m} and some compact abelian group {T}, equipped with a Haar measure {\mu_G} and a measurable homomorphism {\pi\colon O(A_\alpha) \rightarrow G} (using the Loeb {\sigma}-algebra on {O(A_\alpha)} and the Baire {\sigma}-algebra on {G}), with the following properties:

  • (i) {\pi} has dense image, and {\mu_G} is the pushforward of Loeb measure {\mu_{O(A_\alpha)}} by {\pi}.
  • (ii) There exists sets {\{0\} \subset U_0 \subset K_0 \subset G} with {U_0} open and {K_0} compact, such that

    \displaystyle  \pi^{-1}(U_0) \subset 4A_\alpha \subset \pi^{-1}(K_0). \ \ \ \ \ (1)

  • (iii) Whenever {K \subset U \subset G} with {K} compact and {U} open, there exists a nonstandard finite set {B} such that

    \displaystyle  \pi^{-1}(K) \subset B \subset \pi^{-1}(U). \ \ \ \ \ (2)

  • (iv) If {f, g \in L}, then we have the convolution formula

    \displaystyle  f \star g = \pi^*( (\pi_* f) \star (\pi_* g) ) \ \ \ \ \ (3)

    where {\pi_* f,\pi_* g} are the pushforwards of {f,g} to {L^2(G, \mu_G)}, the convolution {\star} on the right-hand side is convolution using {\mu_G}, and {\pi^*} is the pullback map from {L^2(G,\mu_G)} to {L^2(O(A_\alpha), \mu_{O(A_\alpha)})}. In particular, if {\pi_* f = 0}, then {f*g=0} for all {g \in L}.

One can view the locally compact abelian group {G} as a “model “or “Kronecker factor” for the ultra approximate group {A_\alpha} (in close analogy with the Kronecker factor from ergodic theory). In the case that {A_\alpha} is a genuine nonstandard finite group rather than an ultra approximate group, the non-compact components {{\bf R}^d \times {\bf Z}^m} of the Kronecker group {G} are trivial, and this theorem was implicitly established by Szegedy. The compact group {T} is quite large, and in particular is likely to be inseparable; but as with the case of graphons, when one is only studying at most countably many functions {f}, one can cut down the size of this group to be separable (or equivalently, second countable or metrisable) if desired, so one often works with a “reduced Kronecker factor” which is a quotient of the full Kronecker factor {G}. Once one is in the separable case, the Baire sigma algebra is identical with the more familiar Borel sigma algebra.

Given any sequence of uniformly bounded functions {f_n\colon A_n^m \rightarrow {\bf C}} for some fixed {m}, we can view the function {f \in L} defined by

\displaystyle  f := \pi_* \hbox{st} \lim_{n \rightarrow \alpha} f_n \ \ \ \ \ (4)

as an “additive limit” of the {f_n}, in much the same way that graphons {p\colon V_\alpha \times V_\alpha \rightarrow [0,1]} are limits of the indicator functions {1_{E_n}\colon V_n \times V_n \rightarrow \{0,1\}}. The additive limits capture some of the statistics of the {f_n}, for instance the normalised means

\displaystyle  \frac{1}{|A_n|} \sum_{x \in A_n^m} f_n(x)

converge (along the ultrafilter {\alpha}) to the mean

\displaystyle  \int_G f(x)\ d\mu_G(x),

and for three sequences {f_n,g_n,h_n\colon A_n^m \rightarrow {\bf C}} of functions, the normalised correlation

\displaystyle  \frac{1}{|A_n|^2} \sum_{x,y \in A_n^m} f_n(x) g_n(y) h_n(x+y)

converges along {\alpha} to the correlation

\displaystyle  \int_G \int_G f(x) g(y) h(x+y)\ d\mu_G(x) d\mu_G(y),

the normalised {U^2} Gowers norm

\displaystyle  ( \frac{1}{|A_n|^3} \sum_{x,y,z,w \in A_n^m: x+w=y+z} f_n(x) \overline{f_n(y)} \overline{f_n(z)} f_n(w))^{1/4}

converges along {\alpha} to the {U^2} Gowers norm

\displaystyle  ( \int_{G \times G \times G} f(x) \overline{f(y)} \overline{f(z)} f_n(x+y-z)\ d\mu_G(x) d\mu_G(y) d\mu_G(z))^{1/4}

and so forth. We caution however that some correlations that involve evaluating more than one function at the same point will not necessarily be preserved in the additive limit; for instance the normalised {\ell^2} norm

\displaystyle  (\frac{1}{|A_n|} \sum_{x \in A_n^m} |f_n(x)|^2)^{1/2}

does not necessarily converge to the {L^2} norm

\displaystyle  (\int_G |f(x)|^2\ d\mu_G(x))^{1/2},

but can converge instead to a larger quantity, due to the presence of the orthogonal projection {\pi_*} in the definition (4) of {f}.

An important special case of an additive limit occurs when the functions {f_n\colon A_n^m \rightarrow {\bf C}} involved are indicator functions {f_n = 1_{E_n}} of some subsets {E_n} of {A_n^m}. The additive limit {f \in L} does not necessarily remain an indicator function, but instead takes values in {[0,1]} (much as a graphon {p} takes values in {[0,1]} even though the original indicators {1_{E_n}} take values in {\{0,1\}}). The convolution {f \star f\colon G \rightarrow [0,1]} is then the ultralimit of the normalised convolutions {\frac{1}{|A_n|} 1_{E_n} \star 1_{E_n}}; in particular, the measure of the support of {f \star f} provides a lower bound on the limiting normalised cardinality {\frac{1}{|A_n|} |E_n + E_n|} of a sumset. In many situations this lower bound is an equality, but this is not necessarily the case, because the sumset {2E_n = E_n + E_n} could contain a large number of elements which have very few ({o(|A_n|)}) representations as the sum of two elements of {E_n}, and in the limit these portions of the sumset fall outside of the support of {f \star f}. (One can think of the support of {f \star f} as describing the “essential” sumset of {2E_n = E_n + E_n}, discarding those elements that have only very few representations.) Similarly for higher convolutions of {f}. Thus one can use additive limits to partially control the growth {k E_n} of iterated sumsets of subsets {E_n} of approximate groups {A_n}, in the regime where {k} stays bounded and {n} goes to infinity.

Theorem 1 can be proven by Fourier-analytic means (combined with Freiman’s theorem from additive combinatorics), and we will do so below the fold. For now, we give some illustrative examples of additive limits.

Example 2 (Bohr sets) We take {A_n} to be the intervals {A_n := \{ x \in {\bf Z}: |x| \leq N_n \}}, where {N_n} is a sequence going to infinity; these are {2}-approximate groups for all {n}. Let {\theta} be an irrational real number, let {I} be an interval in {{\bf R}/{\bf Z}}, and for each natural number {n} let {B_n} be the Bohr set

\displaystyle  B_n := \{ x \in A^{(n)}: \theta x \hbox{ mod } 1 \in I \}.

In this case, the (reduced) Kronecker factor {G} can be taken to be the infinite cylinder {{\bf R} \times {\bf R}/{\bf Z}} with the usual Lebesgue measure {\mu_G}. The additive limits of {1_{A_n}} and {1_{B_n}} end up being {1_A} and {1_B}, where {A} is the finite cylinder

\displaystyle  A := \{ (x,t) \in {\bf R} \times {\bf R}/{\bf Z}: x \in [-1,1]\}

and {B} is the rectangle

\displaystyle  B := \{ (x,t) \in {\bf R} \times {\bf R}/{\bf Z}: x \in [-1,1]; t \in I \}.

Geometrically, one should think of {A_n} and {B_n} as being wrapped around the cylinder {{\bf R} \times {\bf R}/{\bf Z}} via the homomorphism {x \mapsto (\frac{x}{N_n}, \theta x \hbox{ mod } 1)}, and then one sees that {B_n} is converging in some normalised weak sense to {B}, and similarly for {A_n} and {A}. In particular, the additive limit predicts the growth rate of the iterated sumsets {kB_n} to be quadratic in {k} until {k|I|} becomes comparable to {1}, at which point the growth transitions to linear growth, in the regime where {k} is bounded and {n} is large.

If {\theta = \frac{p}{q}} were rational instead of irrational, then one would need to replace {{\bf R}/{\bf Z}} by the finite subgroup {\frac{1}{q}{\bf Z}/{\bf Z}} here.

Example 3 (Structured subsets of progressions) We take {A_n} be the rank two progression

\displaystyle  A_n := \{ a + b N_n^2: a,b \in {\bf Z}; |a|, |b| \leq N_n \},

where {N_n} is a sequence going to infinity; these are {4}-approximate groups for all {n}. Let {B_n} be the subset

\displaystyle  B_n := \{ a + b N_n^2: a,b \in {\bf Z}; |a|^2 + |b|^2 \leq N_n^2 \}.

Then the (reduced) Kronecker factor can be taken to be {G = {\bf R}^2} with Lebesgue measure {\mu_G}, and the additive limits of the {1_{A_n}} and {1_{B_n}} are then {1_A} and {1_B}, where {A} is the square

\displaystyle  A := \{ (a,b) \in {\bf R}^2: |a|, |b| \leq 1 \}

and {B} is the circle

\displaystyle  B := \{ (a,b) \in {\bf R}^2: a^2+b^2 \leq 1 \}.

Geometrically, the picture is similar to the Bohr set one, except now one uses a Freiman homomorphism {a + b N_n^2 \mapsto (\frac{a}{N_n}, \frac{b}{N_n})} for {a,b = O( N_n )} to embed the original sets {A_n, B_n} into the plane {{\bf R}^2}. In particular, one now expects the growth rate of the iterated sumsets {k A_n} and {k B_n} to be quadratic in {k}, in the regime where {k} is bounded and {n} is large.

Example 4 (Dissociated sets) Let {d} be a fixed natural number, and take

\displaystyle  A_n = \{0, v_1,\dots,v_d,-v_1,\dots,-v_d \}

where {v_1,\dots,v_d} are randomly chosen elements of a large cyclic group {{\bf Z}/p_n{\bf Z}}, where {p_n} is a sequence of primes going to infinity. These are {O(d)}-approximate groups. The (reduced) Kronecker factor {G} can (almost surely) then be taken to be {{\bf Z}^d} with counting measure, and the additive limit of {1_{A_n}} is {1_A}, where {A = \{ 0, e_1,\dots,e_d,-e_1,\dots,-e_d\}} and {e_1,\dots,e_d} is the standard basis of {{\bf Z}^d}. In particular, the growth rates of {k A_n} should grow approximately like {k^d} for {k} bounded and {n} large.

Example 5 (Random subsets of groups) Let {A_n = G_n} be a sequence of finite additive groups whose order is going to infinity. Let {B_n} be a random subset of {G_n} of some fixed density {0 \leq \lambda \leq 1}. Then (almost surely) the Kronecker factor here can be reduced all the way to the trivial group {\{0\}}, and the additive limit of the {1_{B_n}} is the constant function {\lambda}. The convolutions {\frac{1}{|G_n|} 1_{B_n} * 1_{B_n}} then converge in the ultralimit (modulo almost everywhere equivalence) to the pullback of {\lambda^2}; this reflects the fact that {(1-o(1))|G_n|} of the elements of {G_n} can be represented as the sum of two elements of {B_n} in {(\lambda^2 + o(1)) |G_n|} ways. In particular, {B_n+B_n} occupies a proportion {1-o(1)} of {G_n}.

Example 6 (Trigonometric series) Take {A_n = G_n = {\bf Z}/p_n {\bf C}} for a sequence {p_n} of primes going to infinity, and for each {n} let {\xi_{n,1},\xi_{n,2},\dots} be an infinite sequence of frequencies chosen uniformly and independently from {{\bf Z}/p_n{\bf Z}}. Let {f_n\colon {\bf Z}/p_n{\bf Z} \rightarrow {\bf C}} denote the random trigonometric series

\displaystyle  f_n(x) := \sum_{j=1}^\infty 2^{-j} e^{2\pi i \xi_{n,j} x / p_n }.

Then (almost surely) we can take the reduced Kronecker factor {G} to be the infinite torus {({\bf R}/{\bf Z})^{\bf N}} (with the Haar probability measure {\mu_G}), and the additive limit of the {f_n} then becomes the function {f\colon ({\bf R}/{\bf Z})^{\bf N} \rightarrow {\bf R}} defined by the formula

\displaystyle  f( (x_j)_{j=1}^\infty ) := \sum_{j=1}^\infty e^{2\pi i x_j}.

In fact, the pullback {\pi^* f} is the ultralimit of the {f_n}. As such, for any standard exponent {1 \leq q < \infty}, the normalised {l^q} norm

\displaystyle  (\frac{1}{p_n} \sum_{x \in {\bf Z}/p_n{\bf Z}} |f_n(x)|^q)^{1/q}

can be seen to converge to the limit

\displaystyle  (\int_{({\bf R}/{\bf Z})^{\bf N}} |f(x)|^q\ d\mu_G(x))^{1/q}.

The reader is invited to consider combinations of the above examples, e.g. random subsets of Bohr sets, to get a sense of the general case of Theorem 1.

It is likely that this theorem can be extended to the noncommutative setting, using the noncommutative Freiman theorem of Emmanuel Breuillard, Ben Green, and myself, but I have not attempted to do so here (see though this recent preprint of Anush Tserunyan for some related explorations); in a separate direction, there should be extensions that can control higher Gowers norms, in the spirit of the work of Szegedy.

Note: the arguments below will presume some familiarity with additive combinatorics and with nonstandard analysis, and will be a little sketchy in places.

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