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Let us call an arithmetic function ${f: {\bf N} \rightarrow {\bf C}}$ ${1}$-bounded if we have ${|f(n)| \leq 1}$ for all ${n \in {\bf N}}$. In this section we focus on the asymptotic behaviour of ${1}$-bounded multiplicative functions. Some key examples of such functions include:

• The Möbius function ${\mu}$;
• The Liouville function ${\lambda}$;
• Archimedean” characters ${n \mapsto n^{it}}$ (which I call Archimedean because they are pullbacks of a Fourier character ${x \mapsto x^{it}}$ on the multiplicative group ${{\bf R}^+}$, which has the Archimedean property);
• Dirichlet characters (or “non-Archimedean” characters) ${\chi}$ (which are essentially pullbacks of Fourier characters on a multiplicative cyclic group ${({\bf Z}/q{\bf Z})^\times}$ with the discrete (non-Archimedean) metric);
• Hybrid characters ${n \mapsto \chi(n) n^{it}}$.

The space of ${1}$-bounded multiplicative functions is also closed under multiplication and complex conjugation.

Given a multiplicative function ${f}$, we are often interested in the asymptotics of long averages such as

$\displaystyle \frac{1}{X} \sum_{n \leq X} f(n)$

for large values of ${X}$, as well as short sums

$\displaystyle \frac{1}{H} \sum_{x \leq n \leq x+H} f(n)$

where ${H}$ and ${x}$ are both large, but ${H}$ is significantly smaller than ${x}$. (Throughout these notes we will try to normalise most of the sums and integrals appearing here as averages that are trivially bounded by ${O(1)}$; note that other normalisations are preferred in some of the literature cited here.) For instance, as we established in Theorem 58 of Notes 1, the prime number theorem is equivalent to the assertion that

$\displaystyle \frac{1}{X} \sum_{n \leq X} \mu(n) = o(1) \ \ \ \ \ (1)$

as ${X \rightarrow \infty}$. The Liouville function behaves almost identically to the Möbius function, in that estimates for one function almost always imply analogous estimates for the other:

Exercise 1 Without using the prime number theorem, show that (1) is also equivalent to

$\displaystyle \frac{1}{X} \sum_{n \leq X} \lambda(n) = o(1) \ \ \ \ \ (2)$

as ${X \rightarrow \infty}$. (Hint: use the identities ${\lambda(n) = \sum_{d^2|n} \mu(n/d^2)}$ and ${\mu(n) = \sum_{d^2|n} \mu(d) \lambda(n/d^2)}$.)

Henceforth we shall focus our discussion more on the Liouville function, and turn our attention to averages on shorter intervals. From (2) one has

$\displaystyle \frac{1}{H} \sum_{x \leq n \leq x+H} \lambda(n) = o(1) \ \ \ \ \ (3)$

as ${x \rightarrow \infty}$ if ${H = H(x)}$ is such that ${H \geq \varepsilon x}$ for some fixed ${\varepsilon>0}$. However it is significantly more difficult to understand what happens when ${H}$ grows much slower than this. By using the techniques based on zero density estimates discussed in Notes 6, it was shown by Motohashi and that one can also establish \eqref. On the Riemann Hypothesis Maier and Montgomery lowered the threshold to ${H \geq x^{1/2} \log^C x}$ for an absolute constant ${C}$ (the bound ${H \geq x^{1/2+\varepsilon}}$ is more classical, following from Exercise 33 of Notes 2). On the other hand, the randomness heuristics from Supplement 4 suggest that ${H}$ should be able to be taken as small as ${x^\varepsilon}$, and perhaps even ${\log^{1+\varepsilon} x}$ if one is particularly optimistic about the accuracy of these probabilistic models. On the other hand, the Chowla conjecture (mentioned for instance in Supplement 4) predicts that ${H}$ cannot be taken arbitrarily slowly growing in ${x}$, due to the conjectured existence of arbitrarily long strings of consecutive numbers where the Liouville function does not change sign (and in fact one can already show from the known partial results towards the Chowla conjecture that (3) fails for some sequence ${x \rightarrow \infty}$ and some sufficiently slowly growing ${H = H(x)}$, by modifying the arguments in these papers of mine).
The situation is better when one asks to understand the mean value on almost all short intervals, rather than all intervals. There are several equivalent ways to formulate this question:

Exercise 2 Let ${H = H(X)}$ be a function of ${X}$ such that ${H \rightarrow \infty}$ and ${H = o(X)}$ as ${X \rightarrow \infty}$. Let ${f: {\bf N} \rightarrow {\bf C}}$ be a ${1}$-bounded function. Show that the following assertions are equivalent:

• (i) One has

$\displaystyle \frac{1}{H} \sum_{x \leq n \leq x+H} f(n) = o(1)$

as ${X \rightarrow \infty}$, uniformly for all ${x \in [X,2X]}$ outside of a set of measure ${o(X)}$.

• (ii) One has

$\displaystyle \frac{1}{X} \int_X^{2X} |\frac{1}{H} \sum_{x \leq n \leq x+H} f(n)|\ dx = o(1)$

as ${X \rightarrow \infty}$.

• (iii) One has

$\displaystyle \frac{1}{X} \int_X^{2X} |\frac{1}{H} \sum_{x \leq n \leq x+H} f(n)|^2\ dx = o(1) \ \ \ \ \ (4)$

as ${X \rightarrow \infty}$.

As it turns out the second moment formulation in (iii) will be the most convenient for us to work with in this set of notes, as it is well suited to Fourier-analytic techniques (and in particular the Plancherel theorem).

Using zero density methods, for instance, it was shown by Ramachandra that

$\displaystyle \frac{1}{X} \int_X^{2X} |\frac{1}{H} \sum_{x \leq n \leq x+H} \lambda(n)|^2\ dx \ll_{A,\varepsilon} \log^{-A} X$

whenever ${X^{1/6+\varepsilon} \leq H \leq X}$ and ${\varepsilon>0}$. With this quality of bound (saving arbitrary powers of ${\log X}$ over the trivial bound of ${O(1)}$), this is still the lowest value of ${H}$ one can reach unconditionally. However, in a striking recent breakthrough, it was shown by Matomaki and Radziwill that as long as one is willing to settle for weaker bounds (saving a small power of ${\log X}$ or ${\log H}$, or just a qualitative decay of ${o(1)}$), one can obtain non-trivial estimates on far shorter intervals. For instance, they show

Theorem 3 (Matomaki-Radziwill theorem for Liouville) For any ${2 \leq H \leq X}$, one has

$\displaystyle \frac{1}{X} \int_X^{2X} |\frac{1}{H} \sum_{x \leq n \leq x+H} \lambda(n)|^2\ dx \ll \log^{-c} H$

for some absolute constant ${c>0}$.

In fact they prove a slightly more precise result: see Theorem 1 of that paper. In particular, they obtain the asymptotic (4) for any function ${H = H(X)}$ that goes to infinity as ${X \rightarrow \infty}$, no matter how slowly! This ability to let ${H}$ grow slowly with ${X}$ is important for several applications; for instance, in order to combine this type of result with the entropy decrement methods from Notes 9, it is essential that ${H}$ be allowed to grow more slowly than ${\log X}$. See also this survey of Soundararajan for further discussion.

Exercise 4 In this exercise you may use Theorem 3 freely.

• (i) Establish the upper bound

$\displaystyle \frac{1}{X} \sum_{n \leq X} \lambda(n)\lambda(n+1) < 1-c$

for some absolute constant ${c>0}$ and all sufficiently large ${X}$. (Hint: if this bound failed, then ${\lambda(n)=\lambda(n+1)}$ would hold for almost all ${n}$; use this to create many intervals ${[x,x+H]}$ for which ${\frac{1}{H} \sum_{x \leq n \leq x+H} \lambda(n)}$ is extremely large.)

• (ii) Show that Theorem 3 also holds with ${\lambda(n)}$ replaced by ${\chi_2 \lambda(n)}$, where ${\chi_2}$ is the principal character of period ${2}$. (Use the fact that ${\lambda(2n)=-\lambda(n)}$ for all ${n}$.) Use this to establish the corresponding lower bound

$\displaystyle \frac{1}{X} \sum_{n \leq X} \lambda(n)\lambda(n+1) > -1+c$

to (i).

(There is a curious asymmetry to the difficulty level of these bounds; the upper bound in (ii) was established much earlier by Harman, Pintz, and Wolke, but the lower bound in (i) was only established in the Matomaki-Radziwill paper.)

The techniques discussed previously were highly complex-analytic in nature, relying in particular on the fact that functions such as ${\mu}$ or ${\lambda}$ have Dirichlet series ${{\mathcal D} \mu(s) = \frac{1}{\zeta(s)}}$, ${{\mathcal D} \lambda(s) = \frac{\zeta(2s)}{\zeta(s)}}$ that extend meromorphically into the critical strip. In contrast, the Matomaki-Radziwill theorem does not rely on such meromorphic continuations, and in fact holds for more general classes of ${1}$-bounded multiplicative functions ${f}$, for which one typically does not expect any meromorphic continuation into the strip. Instead, one can view the Matomaki-Radziwill theory as following the philosophy of a slightly different approach to multiplicative number theory, namely the pretentious multiplicative number theory of Granville and Soundarajan (as presented for instance in their draft monograph). A basic notion here is the pretentious distance between two ${1}$-bounded multiplicative functions ${f,g}$ (at a given scale ${X}$), which informally measures the extent to which ${f}$ “pretends” to be like ${g}$ (or vice versa). The precise definition is

Definition 5 (Pretentious distance) Given two ${1}$-bounded multiplicative functions ${f,g}$, and a threshold ${X>0}$, the pretentious distance ${\mathbb{D}(f,g;X)}$ between ${f}$ and ${g}$ up to scale ${X}$ is given by the formula

$\displaystyle \mathbb{D}(f,g;X) := \left( \sum_{p \leq X} \frac{1 - \mathrm{Re}(f(p) \overline{g(p)})}{p} \right)^{1/2}$

Note that one can also define an infinite version ${\mathbb{D}(f,g;\infty)}$ of this distance by removing the constraint ${p \leq X}$, though in such cases the pretentious distance may then be infinite. The pretentious distance is not quite a metric (because ${\mathbb{D}(f,f;X)}$ can be non-zero, and furthermore ${\mathbb{D}(f,g;X)}$ can vanish without ${f,g}$ being equal), but it is still quite close to behaving like a metric, in particular it obeys the triangle inequality; see Exercise 16 below. The philosophy of pretentious multiplicative number theory is that two ${1}$-bounded multiplicative functions ${f,g}$ will exhibit similar behaviour at scale ${X}$ if their pretentious distance ${\mathbb{D}(f,g;X)}$ is bounded, but will become uncorrelated from each other if this distance becomes large. A simple example of this philosophy is given by the following “weak Halasz theorem”, proven in Section 2:

Proposition 6 (Logarithmically averaged version of Halasz) Let ${X}$ be sufficiently large. Then for any ${1}$-bounded multiplicative functions ${f,g}$, one has

$\displaystyle \frac{1}{\log X} \sum_{n \leq X} \frac{f(n) \overline{g(n)}}{n} \ll \exp( - c \mathbb{D}(f, g;X)^2 )$

for an absolute constant ${c>0}$.

In particular, if ${f}$ does not pretend to be ${1}$, then the logarithmic average ${\frac{1}{\log X} \sum_{n \leq X} \frac{f(n)}{n}}$ will be small. This condition is basically necessary, since of course ${\frac{1}{\log X} \sum_{n \leq X} \frac{1}{n} = 1 + o(1)}$.

If one works with non-logarithmic averages ${\frac{1}{X} \sum_{n \leq X} f(n)}$, then not pretending to be ${1}$ is insufficient to establish decay, as was already observed in Exercise 11 of Notes 1: if ${f}$ is an Archimedean character ${f(n) = n^{it}}$ for some non-zero real ${t}$, then ${\frac{1}{\log X} \sum_{n \leq X} \frac{f(n)}{n}}$ goes to zero as ${X \rightarrow \infty}$ (which is consistent with Proposition 6), but ${\frac{1}{X} \sum_{n \leq X} f(n)}$ does not go to zero. However, this is in some sense the “only” obstruction to these averages decaying to zero, as quantified by the following basic result:

Theorem 7 (Halasz’s theorem) Let ${X}$ be sufficiently large. Then for any ${1}$-bounded multiplicative function ${f}$, one has

$\displaystyle \frac{1}{X} \sum_{n \leq X} f(n) \ll \exp( - c \min_{|t| \leq T} \mathbb{D}(f, n \mapsto n^{it};X)^2 ) + \frac{1}{T}$

for an absolute constant ${c>0}$ and any ${T > 0}$.

Informally, we refer to a ${1}$-bounded multiplicative function as “pretentious’; if it pretends to be a character such as ${n^{it}}$, and “non-pretentious” otherwise. The precise distinction is rather malleable, as the precise class of characters that one views as “obstructions” varies from situation to situation. For instance, in Proposition 6 it is just the trivial character ${1}$ which needs to be considered, but in Theorem 7 it is the characters ${n \mapsto n^{it}}$ with ${|t| \leq T}$. In other contexts one may also need to add Dirichlet characters ${\chi(n)}$ or hybrid characters such as ${\chi(n) n^{it}}$ to the list of characters that one might pretend to be. The division into pretentious and non-pretentious functions in multiplicative number theory is faintly analogous to the division into major and minor arcs in the circle method applied to additive number theory problems; see Notes 8. The Möbius and Liouville functions are model examples of non-pretentious functions; see Exercise 24.

In the contrapositive, Halasz’ theorem can be formulated as the assertion that if one has a large mean

$\displaystyle |\frac{1}{X} \sum_{n \leq X} f(n)| \geq \eta$

for some ${\eta > 0}$, then one has the pretentious property

$\displaystyle \mathbb{D}( f, n \mapsto n^{it}; X ) \ll \sqrt{\log(1/\eta)}$

for some ${t \ll \eta^{-1}}$. This has the flavour of an “inverse theorem”, of the type often found in arithmetic combinatorics.
Among other things, Halasz’s theorem gives yet another proof of the prime number theorem (1); see Section 2.

We now give a version of the Matomaki-Radziwill theorem for general (non-pretentious) multiplicative functions that is formulated in a similar contrapositive (or “inverse theorem”) fashion, though to simplify the presentation we only state a qualitative version that does not give explicit bounds.

Theorem 8 ((Qualitative) Matomaki-Radziwill theorem) Let ${\eta>0}$, and let ${1 \leq H \leq X}$, with ${H}$ sufficiently large depending on ${\eta}$. Suppose that ${f}$ is a ${1}$-bounded multiplicative function such that

$\displaystyle \frac{1}{X} \int_X^{2X} |\frac{1}{H} \sum_{x \leq n \leq x+H} f(n)|^2\ dx \geq \eta^2.$

Then one has

$\displaystyle \mathbb{D}(f, n \mapsto n^{it};X) \ll_\eta 1$

for some ${t \ll_\eta \frac{X}{H}}$.

The condition ${t \ll_\eta \frac{X}{H}}$ is basically optimal, as the following example shows:

Exercise 9 Let ${\varepsilon>0}$ be a sufficiently small constant, and let ${1 \leq H \leq X}$ be such that ${\frac{1}{\varepsilon} \leq H \leq \varepsilon X}$. Let ${f}$ be the Archimedean character ${f(n) = n^{it}}$ for some ${|t| \leq \varepsilon \frac{X}{H}}$. Show that

$\displaystyle \frac{1}{X} \int_X^{2X} |\frac{1}{H} \sum_{x \leq n \leq x+H} f(n)|^2\ dx \asymp 1.$

Combining Theorem 8 with standard non-pretentiousness facts about the Liouville function (see Exercise 24), we recover Theorem 3 (but with a decay rate of only ${o(1)}$ rather than ${\log^{-c} H}$). We refer the reader to the original paper of Matomaki-Radziwill (as well as this followup paper with myself) for the quantitative version of Theorem 8 that is strong enough to recover the full version of Theorem 3, and which can also handle real-valued pretentious functions.

With our current state of knowledge, the only arguments that can establish the full strength of Halasz and Matomaki-Radziwill theorems are Fourier analytic in nature, relating sums involving an arithmetic function ${f}$ with its Dirichlet series

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

which one can view as a discrete Fourier transform of ${f}$ (or more precisely of the measure ${\sum_{n=1}^\infty \frac{f(n)}{n} \delta_{\log n}}$, if one evaluates the Dirichlet series on the right edge ${\{ 1+it: t \in {\bf R} \}}$ of the critical strip). In this aspect, the techniques resemble the complex-analytic methods from Notes 2, but with the key difference that no analytic or meromorphic continuation into the strip is assumed. The key identity that allows us to pass to Dirichlet series is the following variant of Proposition 7 of Notes 2:

Proposition 10 (Parseval type identity) Let ${f,g: {\bf N} \rightarrow {\bf C}}$ be finitely supported arithmetic functions, and let ${\psi: {\bf R} \rightarrow {\bf R}}$ be a Schwartz function. Then

$\displaystyle \sum_{n=1}^\infty \sum_{m=1}^\infty \frac{f(n)}{n} \frac{\overline{g(m)}}{m} \psi(\log n - \log m) = \frac{1}{2\pi} \int_{\bf R} {\mathcal D} f(1+it) \overline{{\mathcal D} g(1+it)} \hat \psi(t)\ dt$

where ${\hat \psi(t) := \int_{\bf R} \psi(u) e^{itu}\ du}$ is the Fourier transform of ${\psi}$. (Note that the finite support of ${f,g}$ and the Schwartz nature of ${\psi,\hat \psi}$ ensure that both sides of the identity are absolutely convergent.)

The restriction that ${f,g}$ be finitely supported will be slightly annoying in places, since most multiplicative functions will fail to be finitely supported, but this technicality can usually be overcome by suitably truncating the multiplicative function, and taking limits if necessary.

Proof: By expanding out the Dirichlet series, it suffices to show that

$\displaystyle \psi(\log n - \log m) = \frac{1}{2\pi} \int_{\bf R} \frac{1}{n^{it}} \frac{1}{m^{-it}} \hat \psi(t)\ dt$

for any natural numbers ${n,m}$. But this follows from the Fourier inversion formula ${\psi(u) = \frac{1}{2\pi} \int_{\bf R} e^{-itu} \hat \psi(t)\ dt}$ applied at ${u = \log n - \log m}$. $\Box$
For applications to Halasz type theorems, one sets ${g(n)}$ equal to the Kronecker delta ${\delta_{n=1}}$, producing weighted integrals of ${{\mathcal D} f(1+it)}$ of “${L^1}$” type. For applications to Matomaki-Radziwill theorems, one instead sets ${f=g}$, and more precisely uses the following corollary of the above proposition, to obtain weighted integrals of ${|{\mathcal D} f(1+it)|^2}$ of “${L^2}$” type:

Exercise 11 (Plancherel type identity) If ${f: {\bf N} \rightarrow {\bf C}}$ is finitely supported, and ${\varphi: {\bf R} \rightarrow {\bf R}}$ is a Schwartz function, establish the identity

$\displaystyle \int_0^\infty |\sum_{n=1}^\infty \frac{f(n)}{n} \varphi(\log n - \log y)|^2 \frac{dy}{y} = \frac{1}{2\pi} \int_{\bf R} |{\mathcal D} f(1+it)|^2 |\hat \varphi(t)|^2\ dt.$

In contrast, information about the non-pretentious nature of a multiplicative function ${f}$ will give “pointwise” or “${L^\infty}$” type control on the Dirichlet series ${{\mathcal D} f(1+it)}$, as is suggested from the Euler product factorisation of ${{\mathcal D} f}$.

It will be convenient to formalise the notion of ${L^1}$, ${L^2}$, and ${L^\infty}$ control of the Dirichlet series ${{\mathcal D} f}$, which as previously mentioned can be viewed as a sort of “Fourier transform” of ${f}$:

Definition 12 (Fourier norms) Let ${f: {\bf N} \rightarrow {\bf C}}$ be finitely supported, and let ${\Omega \subset {\bf R}}$ be a bounded measurable set. We define the Fourier ${L^\infty}$ norm

$\displaystyle \| f\|_{FL^\infty(\Omega)} := \sup_{t \in \Omega} |{\mathcal D} f(1+it)|,$

the Fourier ${L^2}$ norm

$\displaystyle \| f\|_{FL^2(\Omega)} := \left(\int_\Omega |{\mathcal D} f(1+it)|^2\ dt\right)^{1/2},$

and the Fourier ${L^1}$ norm

$\displaystyle \| f\|_{FL^1(\Omega)} := \int_\Omega |{\mathcal D} f(1+it)|\ dt.$

One could more generally define ${FL^p}$ norms for other exponents ${p}$, but we will only need the exponents ${p=1,2,\infty}$ in this current set of notes. It is clear that all the above norms are in fact (semi-)norms on the space of finitely supported arithmetic functions.

As mentioned above, Halasz’s theorem gives good control on the Fourier ${L^\infty}$ norm for restrictions of non-pretentious functions to intervals:

Exercise 13 (Fourier ${L^\infty}$ control via Halasz) Let ${f: {\bf N} \rightarrow {\bf C}}$ be a ${1}$-bounded multiplicative function, let ${I}$ be an interval in ${[C^{-1} X, CX]}$ for some ${X \geq C \geq 1}$, let ${R \geq 1}$, and let ${\Omega \subset {\bf R}}$ be a bounded measurable set. Show that

$\displaystyle \| f 1_I \|_{FL^\infty(\Omega)} \ll_C \exp( - c \min_{t: \mathrm{dist}(t,\Omega) \leq R} \mathbb{D}(f, n \mapsto n^{it};X)^2 ) + \frac{1}{R}.$

(Hint: you will need to use summation by parts (or an equivalent device) to deal with a ${\frac{1}{n}}$ weight.)

Meanwhile, the Plancherel identity in Exercise 11 gives good control on the Fourier ${L^2}$ norm for functions on long intervals (compare with Exercise 2 from Notes 6):

Exercise 14 (${L^2}$ mean value theorem) Let ${T \geq 1}$, and let ${f: {\bf N} \rightarrow {\bf C}}$ be finitely supported. Show that

$\displaystyle \| f \|_{FL^2([-T,T])}^2 \ll \sum_n \frac{1}{n} (\frac{T}{n} \sum_{m: |n-m| \leq n/T} |f(m)|)^2.$

Conclude in particular that if ${f}$ is supported in ${[C^{-1} N, C N]}$ for some ${C \geq 1}$ and ${N \gg T}$, then

$\displaystyle \| f \|_{FL^2([-T,T])}^2 \ll C^{O(1)} \frac{1}{N} \sum_n |f(n)|^2.$

In the simplest case of the logarithmically averaged Halasz theorem (Proposition 6), Fourier ${L^\infty}$ estimates are already sufficient to obtain decent control on the (weighted) Fourier ${L^1}$ type expressions that show up. However, these estimates are not enough by themselves to establish the full Halasz theorem or the Matomaki-Radziwill theorem. To get from Fourier ${L^\infty}$ control to Fourier ${L^1}$ or ${L^2}$ control more efficiently, the key trick is use Hölder’s inequality, which when combined with the basic Dirichlet series identity

$\displaystyle {\mathcal D}(f*g) = ({\mathcal D} f) ({\mathcal D} g)$

gives the inequalities

$\displaystyle \| f*g \|_{FL^1(\Omega)} \leq \|f\|_{FL^2(\Omega)} \|g\|_{FL^2(\Omega)} \ \ \ \ \ (5)$

and

$\displaystyle \| f*g \|_{FL^2(\Omega)} \leq \|f\|_{FL^2(\Omega)} \|g\|_{FL^\infty(\Omega)} \ \ \ \ \ (6)$

The strategy is then to factor (or approximately factor) the original function ${f}$ as a Dirichlet convolution (or average of convolutions) of various components, each of which enjoys reasonably good Fourier ${L^2}$ or ${L^\infty}$ estimates on various regions ${\Omega}$, and then combine them using the Hölder inequalities (5), (6) and the triangle inequality. For instance, to prove Halasz’s theorem, we will split ${f}$ into the Dirichlet convolution of three factors, one of which will be estimated in ${FL^\infty}$ using the non-pretentiousness hypothesis, and the other two being estimated in ${FL^2}$ using Exercise 14. For the Matomaki-Radziwill theorem, one uses a significantly more complicated decomposition of ${f}$ into a variety of Dirichlet convolutions of factors, and also splits up the Fourier domain ${[-T,T]}$ into several subregions depending on whether the Dirichlet series associated to some of these components are large or small. In each region and for each component of these decompositions, all but one of the factors will be estimated in ${FL^\infty}$, and the other in ${FL^2}$; but the precise way in which this is done will vary from component to component. For instance, in some regions a key factor will be small in ${FL^\infty}$ by construction of the region; in other places, the ${FL^\infty}$ control will come from Exercise 13. Similarly, in some regions, satisfactory ${FL^2}$ control is provided by Exercise 14, but in other regions one must instead use “large value” theorems (in the spirit of Proposition 9 from Notes 6), or amplify the power of the standard ${L^2}$ mean value theorems by combining the Dirichlet series with other Dirichlet series that are known to be large in this region.
There are several ways to achieve the desired factorisation. In the case of Halasz’s theorem, we can simply work with a crude version of the Euler product factorisation, dividing the primes into three categories (“small”, “medium”, and “large” primes) and expressing ${f}$ as a triple Dirichlet convolution accordingly. For the Matomaki-Radziwill theorem, one instead exploits the Turan-Kubilius phenomenon (Section 5 of Notes 1, or Lemma 2 of Notes 9)) that for various moderately wide ranges ${[P,Q]}$ of primes, the number of prime divisors of a large number ${n}$ in the range ${[P,Q]}$ is almost always close to ${\log\log Q - \log\log P}$. Thus, if we introduce the arithmetic functions

$\displaystyle w_{[P,Q]}(n) = \frac{1}{\log\log Q - \log\log P} \sum_{P \leq p \leq Q} 1_{n=p} \ \ \ \ \ (7)$

then we have

$\displaystyle 1 \approx 1 * w_{[P,Q]}$

and more generally we have a twisted approximation

$\displaystyle f \approx f * fw_{[P,Q]}$

for multiplicative functions ${f}$. (Actually, for technical reasons it will be convenient to work with a smoothed out version of these functions; see Section 3.) Informally, these formulas suggest that the “${FL^2}$ energy” of a multiplicative function ${f}$ is concentrated in those regions where ${f w_{[P,Q]}}$ is extremely large in a ${FL^\infty}$ sense. Iterations of this formula (or variants of this formula, such as an identity due to Ramaré) will then give the desired (approximate) factorisation of ${{\mathcal D} f}$.
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In these notes we presume familiarity with the basic concepts of probability theory, such as random variables (which could take values in the reals, vectors, or other measurable spaces), probability, and expectation. Much of this theory is in turn based on measure theory, which we will also presume familiarity with. See for instance this previous set of lecture notes for a brief review.

The basic objects of study in analytic number theory are deterministic; there is nothing inherently random about the set of prime numbers, for instance. Despite this, one can still interpret many of the averages encountered in analytic number theory in probabilistic terms, by introducing random variables into the subject. Consider for instance the form

$\displaystyle \sum_{n \leq x} \mu(n) = o(x) \ \ \ \ \ (1)$

of the prime number theorem (where we take the limit ${x \rightarrow \infty}$). One can interpret this estimate probabilistically as

$\displaystyle {\mathbb E} \mu(\mathbf{n}) = o(1) \ \ \ \ \ (2)$

where ${\mathbf{n} = \mathbf{n}_{\leq x}}$ is a random variable drawn uniformly from the natural numbers up to ${x}$, and ${{\mathbb E}}$ denotes the expectation. (In this set of notes we will use boldface symbols to denote random variables, and non-boldface symbols for deterministic objects.) By itself, such an interpretation is little more than a change of notation. However, the power of this interpretation becomes more apparent when one then imports concepts from probability theory (together with all their attendant intuitions and tools), such as independence, conditioning, stationarity, total variation distance, and entropy. For instance, suppose we want to use the prime number theorem (1) to make a prediction for the sum

$\displaystyle \sum_{n \leq x} \mu(n) \mu(n+1).$

After dividing by ${x}$, this is essentially

$\displaystyle {\mathbb E} \mu(\mathbf{n}) \mu(\mathbf{n}+1).$

With probabilistic intuition, one may expect the random variables ${\mu(\mathbf{n}), \mu(\mathbf{n}+1)}$ to be approximately independent (there is no obvious relationship between the number of prime factors of ${\mathbf{n}}$, and of ${\mathbf{n}+1}$), and so the above average would be expected to be approximately equal to

$\displaystyle ({\mathbb E} \mu(\mathbf{n})) ({\mathbb E} \mu(\mathbf{n}+1))$

which by (2) is equal to ${o(1)}$. Thus we are led to the prediction

$\displaystyle \sum_{n \leq x} \mu(n) \mu(n+1) = o(x). \ \ \ \ \ (3)$

The asymptotic (3) is widely believed (it is a special case of the Chowla conjecture, which we will discuss in later notes; while there has been recent progress towards establishing it rigorously, it remains open for now.
How would one try to make these probabilistic intuitions more rigorous? The first thing one needs to do is find a more quantitative measurement of what it means for two random variables to be “approximately” independent. There are several candidates for such measurements, but we will focus in these notes on two particularly convenient measures of approximate independence: the “${L^2}$” measure of independence known as covariance, and the “${L \log L}$” measure of independence known as mutual information (actually we will usually need the more general notion of conditional mutual information that measures conditional independence). The use of ${L^2}$ type methods in analytic number theory is well established, though it is usually not described in probabilistic terms, being referred to instead by such names as the “second moment method”, the “large sieve” or the “method of bilinear sums”. The use of ${L \log L}$ methods (or “entropy methods”) is much more recent, and has been able to control certain types of averages in analytic number theory that were out of reach of previous methods such as ${L^2}$ methods. For instance, in later notes we will use entropy methods to establish the logarithmically averaged version

$\displaystyle \sum_{n \leq x} \frac{\mu(n) \mu(n+1)}{n} = o(\log x) \ \ \ \ \ (4)$

of (3), which is implied by (3) but strictly weaker (much as the prime number theorem (1) implies the bound ${\sum_{n \leq x} \frac{\mu(n)}{n} = o(\log x)}$, but the latter bound is much easier to establish than the former).
As with many other situations in analytic number theory, we can exploit the fact that certain assertions (such as approximate independence) can become significantly easier to prove if one only seeks to establish them on average, rather than uniformly. For instance, given two random variables ${\mathbf{X}}$ and ${\mathbf{Y}}$ of number-theoretic origin (such as the random variables ${\mu(\mathbf{n})}$ and ${\mu(\mathbf{n}+1)}$ mentioned previously), it can often be extremely difficult to determine the extent to which ${\mathbf{X},\mathbf{Y}}$ behave “independently” (or “conditionally independently”). However, thanks to second moment tools or entropy based tools, it is often possible to assert results of the following flavour: if ${\mathbf{Y}_1,\dots,\mathbf{Y}_k}$ are a large collection of “independent” random variables, and ${\mathbf{X}}$ is a further random variable that is “not too large” in some sense, then ${\mathbf{X}}$ must necessarily be nearly independent (or conditionally independent) to many of the ${\mathbf{Y}_i}$, even if one cannot pinpoint precisely which of the ${\mathbf{Y}_i}$ the variable ${\mathbf{X}}$ is independent with. In the case of the second moment method, this allows us to compute correlations such as ${{\mathbb E} {\mathbf X} \mathbf{Y}_i}$ for “most” ${i}$. The entropy method gives bounds that are significantly weaker quantitatively than the second moment method (and in particular, in its current incarnation at least it is only able to say non-trivial assertions involving interactions with residue classes at small primes), but can control significantly more general quantities ${{\mathbb E} F( {\mathbf X}, \mathbf{Y}_i )}$ for “most” ${i}$ thanks to tools such as the Pinsker inequality.

In the fall quarter (starting Sep 27) I will be teaching a graduate course on analytic prime number theory.  This will be similar to a graduate course I taught in 2015, and in particular will reuse several of the lecture notes from that course, though it will also incorporate some new material (and omit some material covered in the previous course, to compensate).  I anticipate covering the following topics:

1. Elementary multiplicative number theory
2. Complex-analytic multiplicative number theory
3. The entropy decrement argument
4. Bounds for exponential sums
5. Zero density theorems
6. Halasz’s theorem and the Matomaki-Radziwill theorem
7. The circle method
8. (If time permits) Chowla’s conjecture and the Erdos discrepancy problem [Update: I did not end up writing notes on this topic.]

Lecture notes for topics 3, 6, and 8 will be forthcoming.

We have seen in previous notes that the operation of forming a Dirichlet series

$\displaystyle {\mathcal D} f(n) := \sum_n \frac{f(n)}{n^s}$

or twisted Dirichlet series

$\displaystyle {\mathcal D} \chi f(n) := \sum_n \frac{f(n) \chi(n)}{n^s}$

is an incredibly useful tool for questions in multiplicative number theory. Such series can be viewed as a multiplicative Fourier transform, since the functions ${n \mapsto \frac{1}{n^s}}$ and ${n \mapsto \frac{\chi(n)}{n^s}}$ are multiplicative characters.
Similarly, it turns out that the operation of forming an additive Fourier series

$\displaystyle \hat f(\theta) := \sum_n f(n) e(-n \theta),$

where ${\theta}$ lies on the (additive) unit circle ${{\bf R}/{\bf Z}}$ and ${e(\theta) := e^{2\pi i \theta}}$ is the standard additive character, is an incredibly useful tool for additive number theory, particularly when studying additive problems involving three or more variables taking values in sets such as the primes; the deployment of this tool is generally known as the Hardy-Littlewood circle method. (In the analytic number theory literature, the minus sign in the phase ${e(-n\theta)}$ is traditionally omitted, and what is denoted by ${\hat f(\theta)}$ here would be referred to instead by ${S_f(-\theta)}$, ${S(f;-\theta)}$ or just ${S(-\theta)}$.) We list some of the most classical problems in this area:

• (Even Goldbach conjecture) Is it true that every even natural number ${N}$ greater than two can be expressed as the sum ${p_1+p_2}$ of two primes?
• (Odd Goldbach conjecture) Is it true that every odd natural number ${N}$ greater than five can be expressed as the sum ${p_1+p_2+p_3}$ of three primes?
• (Waring problem) For each natural number ${k}$, what is the least natural number ${g(k)}$ such that every natural number ${N}$ can be expressed as the sum of ${g(k)}$ or fewer ${k^{th}}$ powers?
• (Asymptotic Waring problem) For each natural number ${k}$, what is the least natural number ${G(k)}$ such that every sufficiently large natural number ${N}$ can be expressed as the sum of ${G(k)}$ or fewer ${k^{th}}$ powers?
• (Partition function problem) For any natural number ${N}$, let ${p(N)}$ denote the number of representations of ${N}$ of the form ${N = n_1 + \dots + n_k}$ where ${k}$ and ${n_1 \geq \dots \geq n_k}$ are natural numbers. What is the asymptotic behaviour of ${p(N)}$ as ${N \rightarrow \infty}$?

The Waring problem and its asymptotic version will not be discussed further here, save to note that the Vinogradov mean value theorem (Theorem 13 from Notes 5) and its variants are particularly useful for getting good bounds on ${G(k)}$; see for instance the ICM article of Wooley for recent progress on these problems. Similarly, the partition function problem was the original motivation of Hardy and Littlewood in introducing the circle method, but we will not discuss it further here; see e.g. Chapter 20 of Iwaniec-Kowalski for a treatment.

Instead, we will focus our attention on the odd Goldbach conjecture as our model problem. (The even Goldbach conjecture, which involves only two variables instead of three, is unfortunately not amenable to a circle method approach for a variety of reasons, unless the statement is replaced with something weaker, such as an averaged statement; see this previous blog post for further discussion. On the other hand, the methods here can obtain weaker versions of the even Goldbach conjecture, such as showing that “almost all” even numbers are the sum of two primes; see Exercise 34 below.) In particular, we will establish the following celebrated theorem of Vinogradov:

Theorem 1 (Vinogradov’s theorem) Every sufficiently large odd number ${N}$ is expressible as the sum of three primes.

Recently, the restriction that ${n}$ be sufficiently large was replaced by Helfgott with ${N > 5}$, thus establishing the odd Goldbach conjecture in full. This argument followed the same basic approach as Vinogradov (based on the circle method), but with various estimates replaced by “log-free” versions (analogous to the log-free zero-density theorems in Notes 7), combined with careful numerical optimisation of constants and also some numerical work on the even Goldbach problem and on the generalised Riemann hypothesis. We refer the reader to Helfgott’s text for details.

We will in fact show the more precise statement:

Theorem 2 (Quantitative Vinogradov theorem) Let ${N \geq 2}$ be an natural number. Then

$\displaystyle \sum_{a,b,c: a+b+c=N} \Lambda(a) \Lambda(b) \Lambda(c) = G_3(N) \frac{N^2}{2} + O_A( N^2 \log^{-A} N )$

for any ${A>0}$, where

$\displaystyle G_3(N) = \prod_{p|N} (1-\frac{1}{(p-1)^2}) \times \prod_{p \not | N} (1 + \frac{1}{(p-1)^3}). \ \ \ \ \ (1)$

The implied constants are ineffective.

We dropped the hypothesis that ${N}$ is odd in Theorem 2, but note that ${G_3(N)}$ vanishes when ${N}$ is even. For odd ${N}$, we have

$\displaystyle 1 \ll G_3(N) \ll 1.$

Exercise 3 Show that Theorem 2 implies Theorem 1.

Unfortunately, due to the ineffectivity of the constants in Theorem 2 (a consequence of the reliance on the Siegel-Walfisz theorem in the proof of that theorem), one cannot quantify explicitly what “sufficiently large” means in Theorem 1 directly from Theorem 2. However, there is a modification of this theorem which gives effective bounds; see Exercise 32 below.

Exercise 4 Obtain a heuristic derivation of the main term ${G_3(N) \frac{N^2}{2}}$ using the modified Cramér model (Section 1 of Supplement 4).

To prove Theorem 2, we consider the more general problem of estimating sums of the form

$\displaystyle \sum_{a,b,c \in {\bf Z}: a+b+c=N} f(a) g(b) h(c)$

for various integers ${N}$ and functions ${f,g,h: {\bf Z} \rightarrow {\bf C}}$, which we will take to be finitely supported to avoid issues of convergence.
Suppose that ${f,g,h}$ are supported on ${\{1,\dots,N\}}$; for simplicity, let us first assume the pointwise bound ${|f(n)|, |g(n)|, |h(n)| \ll 1}$ for all ${n}$. (This simple case will not cover the case in Theorem 2, when ${f,g,h}$ are truncated versions of the von Mangoldt function ${\Lambda}$, but will serve as a warmup to that case.) Then we have the trivial upper bound

$\displaystyle \sum_{a,b,c \in {\bf Z}: a+b+c=N} f(a) g(b) h(c) \ll N^2. \ \ \ \ \ (2)$

A basic observation is that this upper bound is attainable if ${f,g,h}$ all “pretend” to behave like the same additive character ${n \mapsto e(\theta n)}$ for some ${\theta \in {\bf R}/{\bf Z}}$. For instance, if ${f(n)=g(n)=h(n) = e(\theta n) 1_{n \leq N}}$, then we have ${f(a)g(b)h(c) = e(\theta N)}$ when ${a+b+c=N}$, and then it is not difficult to show that

$\displaystyle \sum_{a,b,c \in {\bf Z}: a+b+c=N} f(a) g(b) h(c) = (\frac{1}{2}+o(1)) e(\theta N) N^2$

as ${N \rightarrow \infty}$.
The key to the success of the circle method lies in the converse of the above statement: the only way that the trivial upper bound (2) comes close to being sharp is when ${f,g,h}$ all correlate with the same character ${n \mapsto e(\theta n)}$, or in other words ${\hat f(\theta), \hat g(\theta), \hat h(\theta)}$ are simultaneously large. This converse is largely captured by the following two identities:

Exercise 5 Let ${f,g,h: {\bf Z} \rightarrow {\bf C}}$ be finitely supported functions. Then for any natural number ${N}$, show that

$\displaystyle \sum_{a,b,c: a+b+c=N} f(a) g(b) h(c) = \int_{{\bf R}/{\bf Z}} \hat f(\theta) \hat g(\theta) \hat h(\theta) e(\theta N)\ d\theta \ \ \ \ \ (3)$

and

$\displaystyle \sum_n |f(n)|^2 = \int_{{\bf R}/{\bf Z}} |\hat f(\theta)|^2\ d\theta.$

The traditional approach to using the circle method to compute sums such as ${\sum_{a,b,c: a+b+c=N} f(a) g(b) h(c)}$ proceeds by invoking (3) to express this sum as an integral over the unit circle, then dividing the unit circle into “major arcs” where ${\hat f(\theta), \hat g(\theta),\hat h(\theta)}$ are large but computable with high precision, and “minor arcs” where one has estimates to ensure that ${\hat f(\theta), \hat g(\theta),\hat h(\theta)}$ are small in both ${L^\infty}$ and ${L^2}$ senses. For functions ${f,g,h}$ of number-theoretic significance, such as truncated von Mangoldt functions, the “major arcs” typically consist of those ${\theta}$ that are close to a rational number ${\frac{a}{q}}$ with ${q}$ not too large, and the “minor arcs” consist of the remaining portions of the circle. One then obtains lower bounds on the contributions of the major arcs, and upper bounds on the contribution of the minor arcs, in order to get good lower bounds on ${\sum_{a,b,c: a+b+c=N} f(a) g(b) h(c)}$.

This traditional approach is covered in many places, such as this text of Vaughan. We will emphasise in this set of notes a slightly different perspective on the circle method, coming from recent developments in additive combinatorics; this approach does not quite give the sharpest quantitative estimates, but it allows for easier generalisation to more combinatorial contexts, for instance when replacing the primes by dense subsets of the primes, or replacing the equation ${a+b+c=N}$ with some other equation or system of equations.

From Exercise 5 and Hölder’s inequality, we immediately obtain

Corollary 6 Let ${f,g,h: {\bf Z} \rightarrow {\bf C}}$ be finitely supported functions. Then for any natural number ${N}$, we have

$\displaystyle |\sum_{a,b,c: a+b+c=N} f(a) g(b) h(c)| \leq (\sum_n |f(n)|^2)^{1/2} (\sum_n |g(n)|^2)^{1/2}$

$\displaystyle \times \sup_\theta |\sum_n h(n) e(n\theta)|.$

Similarly for permutations of the ${f,g,h}$.

In the case when ${f,g,h}$ are supported on ${[1,N]}$ and bounded by ${O(1)}$, this corollary tells us that we have ${\sum_{a,b,c: a+b+c=N} f(a) g(b) h(c)}$ is ${o(N^2)}$ whenever one has ${\sum_n h(n) e(n\theta) = o(N)}$ uniformly in ${\theta}$, and similarly for permutations of ${f,g,h}$. From this and the triangle inequality, we obtain the following conclusion: if ${f}$ is supported on ${[1,N]}$ and bounded by ${O(1)}$, and ${f}$ is Fourier-approximated by another function ${g}$ supported on ${[1,N]}$ and bounded by ${O(1)}$ in the sense that

$\displaystyle \sum_n f(n) e(n\theta) = \sum_n g(n) e(n\theta) + o(N)$

uniformly in ${\theta}$, then we have

$\displaystyle \sum_{a,b,c: a+b+c=N} f(a) f(b) f(c) = \sum_{a,b,c: a+b+c=N} g(a) g(b) g(c) + o(N^2). \ \ \ \ \ (4)$

Thus, one possible strategy for estimating the sum ${\sum_{a,b,c: a+b+c=N} f(a) f(b) f(c)}$ is, one can effectively replace (or “model”) ${f}$ by a simpler function ${g}$ which Fourier-approximates ${g}$ in the sense that the exponential sums ${\sum_n f(n) e(n\theta), \sum_n g(n) e(n\theta)}$ agree up to error ${o(N)}$. For instance:

Exercise 7 Let ${N}$ be a natural number, and let ${A}$ be a random subset of ${\{1,\dots,N\}}$, chosen so that each ${n \in \{1,\dots,N\}}$ has an independent probability of ${1/2}$ of lying in ${A}$.

• (i) If ${f := 1_A}$ and ${g := \frac{1}{2} 1_{[1,N]}}$, show that with probability ${1-o(1)}$ as ${N \rightarrow \infty}$, one has ${\sum_n f(n) e(n\theta) = \sum_n g(n) e(n\theta) + o(N)}$ uniformly in ${\theta}$. (Hint: for any fixed ${\theta}$, this can be accomplished with quite a good probability (e.g. ${1-o(N^{-2})}$) using a concentration of measure inequality, such as Hoeffding’s inequality. To obtain the uniformity in ${\theta}$, round ${\theta}$ to the nearest multiple of (say) ${1/N^2}$ and apply the union bound).
• (ii) Show that with probability ${1-o(1)}$, one has ${(\frac{1}{16}+o(1))N^2}$ representations of the form ${N=a+b+c}$ with ${a,b,c \in A}$ (with ${(a,b,c)}$ treated as an ordered triple, rather than an unordered one).

In the case when ${f}$ is something like the truncated von Mangoldt function ${\Lambda(n) 1_{n \leq N}}$, the quantity ${\sum_n |f(n)|^2}$ is of size ${O( N \log N)}$ rather than ${O( N )}$. This costs us a logarithmic factor in the above analysis, however we can still conclude that we have the approximation (4) whenever ${g}$ is another sequence with ${\sum_n |g(n)|^2 \ll N \log N}$ such that one has the improved Fourier approximation

$\displaystyle \sum_n f(n) e(n\theta) = \sum_n g(n) e(n\theta) + o(\frac{N}{\log N}) \ \ \ \ \ (5)$

uniformly in ${\theta}$. (Later on we will obtain a “log-free” version of this implication in which one does not need to gain a factor of ${\frac{1}{\log N}}$ in the error term.)
This suggests a strategy for proving Vinogradov’s theorem: find an approximant ${g}$ to some suitable truncation ${f}$ of the von Mangoldt function (e.g. ${f(n) = \Lambda(n) 1_{n \leq N}}$ or ${f(n) = \Lambda(n) \eta(n/N)}$) which obeys the Fourier approximation property (5), and such that the expression ${\sum_{a+b+c=N} g(a) g(b) g(c)}$ is easily computable. It turns out that there are a number of good options for such an approximant ${g}$. One of the quickest ways to obtain such an approximation (which is used in Chapter 19 of Iwaniec and Kowalski) is to start with the standard identity ${\Lambda = -\mu L * 1}$, that is to say

$\displaystyle \Lambda(n) = - \sum_{d|n} \mu(d) \log d,$

and obtain an approximation by truncating ${d}$ to be less than some threshold ${R}$ (which, in practice, would be a small power of ${N}$):

$\displaystyle \Lambda(n) \approx - \sum_{d \leq R: d|n} \mu(d) \log d. \ \ \ \ \ (6)$

Thus, for instance, if ${f(n) = \Lambda(n) 1_{n \leq N}}$, the approximant ${g}$ would be taken to be

$\displaystyle g(n) := - \sum_{d \leq R: d|n} \mu(d) \log d 1_{n \leq N}.$

One could also use the slightly smoother approximation

$\displaystyle \Lambda(n) \approx \sum_{d \leq R: d|n} \mu(d) \log \frac{R}{d} \ \ \ \ \ (7)$

in which case we would take

$\displaystyle g(n) := \sum_{d \leq R: d|n} \mu(d) \log \frac{R}{d} 1_{n \leq N}.$

The function ${g}$ is somewhat similar to the continuous Selberg sieve weights studied in Notes 4, with the main difference being that we did not square the divisor sum as we will not need to take ${g}$ to be non-negative. As long as ${z}$ is not too large, one can use some sieve-like computations to compute expressions like ${\sum_{a+b+c=N} g(a)g(b)g(c)}$ quite accurately. The approximation (5) can be justified by using a nice estimate of Davenport that exemplifies the Mobius pseudorandomness heuristic from Supplement 4:

Theorem 8 (Davenport’s estimate) For any ${A>0}$ and ${x \geq 2}$, we have

$\displaystyle \sum_{n \leq x} \mu(n) e(\theta n) \ll_A x \log^{-A} x$

uniformly for all ${\theta \in {\bf R}/{\bf Z}}$. The implied constants are ineffective.

This estimate will be proven by splitting into two cases. In the “major arc” case when ${\theta}$ is close to a rational ${a/q}$ with ${q}$ small (of size ${O(\log^{O(1)} x)}$ or so), this estimate will be a consequence of the Siegel-Walfisz theorem ( from Notes 2); it is the application of this theorem that is responsible for the ineffective constants. In the remaining “minor arc” case, one proceeds by using a combinatorial identity (such as Vaughan’s identity) to express the sum ${\sum_{n \leq x} \mu(n) e(\theta n)}$ in terms of bilinear sums of the form ${\sum_n \sum_m a_n b_m e(\theta nm)}$, and use the Cauchy-Schwarz inequality and the minor arc nature of ${\theta}$ to obtain a gain in this case. This will all be done below the fold. We will also use (a rigorous version of) the approximation (6) (or (7)) to establish Vinogradov’s theorem.

A somewhat different looking approximation for the von Mangoldt function that also turns out to be quite useful is

$\displaystyle \Lambda(n) \approx \sum_{q \leq Q} \sum_{a \in ({\bf Z}/q{\bf Z})^\times} \frac{\mu(q)}{\phi(q)} e( \frac{an}{q} ) \ \ \ \ \ (8)$

for some ${Q}$ that is not too large compared to ${N}$. The methods used to establish Theorem 8 can also establish a Fourier approximation that makes (8) precise, and which can yield an alternate proof of Vinogradov’s theorem; this will be done below the fold.
The approximation (8) can be written in a way that makes it more similar to (7):

Exercise 9 Show that the right-hand side of (8) can be rewritten as

$\displaystyle \sum_{d \leq Q: d|n} \mu(d) \rho_d$

where

$\displaystyle \rho_d := \frac{d}{\phi(d)} \sum_{m \leq Q/d: (m,d)=1} \frac{\mu^2(m)}{\phi(m)}.$

Then, show the inequalities

$\displaystyle \sum_{m \leq Q/d} \frac{\mu^2(m)}{\phi(m)} \leq \rho_d \leq \sum_{m \leq Q} \frac{\mu^2(m)}{\phi(m)}$

and conclude that

$\displaystyle \log \frac{Q}{d} - O(1) \leq \rho_d \leq \log Q + O(1).$

(Hint: for the latter estimate, use Theorem 27 of Notes 1.)

The coefficients ${\rho_d}$ in the above exercise are quite similar to optimised Selberg sieve coefficients (see Section 2 of Notes 4).

Another approximation to ${\Lambda}$, related to the modified Cramér random model (see Model 10 of Supplement 4) is

$\displaystyle \Lambda(n) \approx \frac{W}{\phi(W)} 1_{(n,W)=1} \ \ \ \ \ (9)$

where ${W := \prod_{p \leq w} p}$ and ${w}$ is a slowly growing function of ${N}$ (e.g. ${w = \log\log N}$); a closely related approximation is

$\displaystyle \frac{\phi(W)}{W} \Lambda(Wn+b) \approx 1 \ \ \ \ \ (10)$

for ${W,w}$ as above and ${1 \leq b \leq W}$ coprime to ${W}$. These approximations (closely related to a device known as the “${W}$-trick”) are not as quantitatively accurate as the previous approximations, but can still suffice to establish Vinogradov’s theorem, and also to count many other linear patterns in the primes or subsets of the primes (particularly if one injects some additional tools from additive combinatorics, and specifically the inverse conjecture for the Gowers uniformity norms); see this paper of Ben Green and myself for more discussion (and this more recent paper of Shao for an analysis of this approach in the context of Vinogradov-type theorems). The following exercise expresses the approximation (9) in a form similar to the previous approximation (8):

Exercise 10 With ${W}$ as above, show that

$\displaystyle \frac{W}{\phi(W)} 1_{(n,W)=1} = \sum_{q|W} \sum_{a \in ({\bf Z}/q{\bf Z})^\times} \frac{\mu(q)}{\phi(q)} e( \frac{an}{q} )$

for all natural numbers ${n}$.

A major topic of interest of analytic number theory is the asymptotic behaviour of the Riemann zeta function ${\zeta}$ in the critical strip ${\{ \sigma+it: 0 < \sigma < 1; t \in {\bf R} \}}$ in the limit ${t \rightarrow +\infty}$. For the purposes of this set of notes, it is a little simpler technically to work with the log-magnitude ${\log |\zeta|: {\bf C} \rightarrow [-\infty,+\infty]}$ of the zeta function. (In principle, one can reconstruct a branch of ${\log \zeta}$, and hence ${\zeta}$ itself, from ${\log |\zeta|}$ using the Cauchy-Riemann equations, or tools such as the Borel-Carathéodory theorem, see Exercise 40 of Supplement 2.)

One has the classical estimate

$\displaystyle \zeta(\sigma+it) = O( t^{O(1)} )$

when ${\sigma = O(1)}$ and ${t \geq 10}$ (say), so that

$\displaystyle \log |\zeta(\sigma+it)| \leq O( \log t ). \ \ \ \ \ (1)$

(See e.g. Exercise 37 from Supplement 3.) In view of this, let us define the normalised log-magnitudes ${F_T: {\bf C} \rightarrow [-\infty,+\infty]}$ for any ${T \geq 10}$ by the formula

$\displaystyle F_T( \sigma + it ) := \frac{1}{\log T} \log |\zeta( \sigma + i(T + t) )|;$

informally, this is a normalised window into ${\log |\zeta|}$ near ${iT}$. One can rephrase several assertions about the zeta function in terms of the asymptotic behaviour of ${F_T}$. For instance:

• (i) The bound (1) implies that ${F_T}$ is asymptotically locally bounded from above in the limit ${T \rightarrow \infty}$, thus for any compact set ${K \subset {\bf C}}$ we have ${F_T(\sigma+it) \leq O_K(1)}$ for ${\sigma+it \in K}$ and ${T}$ sufficiently large. In fact the implied constant in ${K}$ only depends on the projection of ${K}$ to the real axis.
• (ii) For ${\sigma > 1}$, we have the bounds

$\displaystyle |\zeta(\sigma+it)|, \frac{1}{|\zeta(\sigma+it)|} \leq \zeta(\sigma)$

which implies that ${F_T}$ converges locally uniformly as ${T \rightarrow +\infty}$ to zero in the region ${\{ \sigma+it: \sigma > 1, t \in {\bf R} \}}$.

• (iii) The functional equation, together with the symmetry ${\zeta(\sigma-it) = \overline{\zeta(\sigma+it)}}$, implies that

$\displaystyle |\zeta(\sigma+it)| = 2^\sigma \pi^{\sigma-1} |\sin \frac{\pi(\sigma+it)}{2}| |\Gamma(1-\sigma-it)| |\zeta(1-\sigma+it)|$

which by Exercise 17 of Supplement 3 shows that

$\displaystyle F_T( 1-\sigma+it ) = \frac{1}{2}-\sigma + F_T(\sigma+it) + o(1)$

as ${T \rightarrow \infty}$, locally uniformly in ${\sigma+it}$. In particular, when combined with the previous item, we see that ${F_T(\sigma+it)}$ converges locally uniformly as ${T \rightarrow +\infty}$ to ${\frac{1}{2}-\sigma}$ in the region ${\{ \sigma+it: \sigma < 0, t \in {\bf R}\}}$.

• (iv) From Jensen’s formula (Theorem 16 of Supplement 2) we see that ${\log|\zeta|}$ is a subharmonic function, and thus ${F_T}$ is subharmonic as well. In particular we have the mean value inequality

$\displaystyle F_T( z_0 ) \leq \frac{1}{\pi r^2} \int_{z: |z-z_0| \leq r} F_T(z)$

for any disk ${\{ z: |z-z_0| \leq r \}}$, where the integral is with respect to area measure. From this and (ii) we conclude that

$\displaystyle \int_{z: |z-z_0| \leq r} F_T(z) \geq O_{z_0,r}(1)$

for any disk with ${\hbox{Re}(z_0)>1}$ and sufficiently large ${T}$; combining this with (i) we conclude that ${F_T}$ is asymptotically locally bounded in ${L^1}$ in the limit ${T \rightarrow \infty}$, thus for any compact set ${K \subset {\bf C}}$ we have ${\int_K |F_T| \ll_K 1}$ for sufficiently large ${T}$.

From (iv) and the usual Arzela-Ascoli diagonalisation argument, we see that the ${F_T}$ are asymptotically compact in the topology of distributions: given any sequence ${T_n}$ tending to ${+\infty}$, one can extract a subsequence such that the ${F_T}$ converge in the sense of distributions. Let us then define a normalised limit profile of ${\log|\zeta|}$ to be a distributional limit ${F}$ of a sequence of ${F_T}$; they are analogous to limiting profiles in PDE, and also to the more recent introduction of “graphons” in the theory of graph limits. Then by taking limits in (i)-(iv) we can say a lot about such normalised limit profiles ${F}$ (up to almost everywhere equivalence, which is an issue we will address shortly):

• (i) ${F}$ is bounded from above in the critical strip ${\{ \sigma+it: 0 \leq \sigma \leq 1 \}}$.
• (ii) ${F}$ vanishes on ${\{ \sigma+it: \sigma \geq 1\}}$.
• (iii) We have the functional equation ${F(1-\sigma+it) = \frac{1}{2}-\sigma + F(\sigma+it)}$ for all ${\sigma+it}$. In particular ${F(\sigma+it) = \frac{1}{2}-\sigma}$ for ${\sigma<0}$.
• (iv) ${F}$ is subharmonic.

Unfortunately, (i)-(iv) fail to characterise ${F}$ completely. For instance, one could have ${F(\sigma+it) = f(\sigma)}$ for any convex function ${f(\sigma)}$ of ${\sigma}$ that equals ${0}$ for ${\sigma \geq 1}$, ${\frac{1}{2}-\sigma}$ for ${\sigma \leq 1}$, and obeys the functional equation ${f(1-\sigma) = \frac{1}{2}-\sigma+f(\sigma)}$, and this would be consistent with (i)-(iv). One can also perturb such examples in a region where ${f}$ is strictly convex to create further examples of functions obeying (i)-(iv). Note from subharmonicity that the function ${\sigma \mapsto \sup_t F(\sigma+it)}$ is always going to be convex in ${\sigma}$; this can be seen as a limiting case of the Hadamard three-lines theorem (Exercise 41 of Supplement 2).

We pause to address one minor technicality. We have defined ${F}$ as a distributional limit, and as such it is a priori only defined up to almost everywhere equivalence. However, due to subharmonicity, there is a unique upper semi-continuous representative of ${F}$ (taking values in ${[-\infty,+\infty)}$), defined by the formula

$\displaystyle F(z_0) = \lim_{r \rightarrow 0^+} \frac{1}{\pi r^2} \int_{B(z_0,r)} F(z)\ dz$

for any ${z_0 \in {\bf C}}$ (note from subharmonicity that the expression in the limit is monotone nonincreasing as ${r \rightarrow 0}$, and is also continuous in ${z_0}$). We will now view this upper semi-continuous representative of ${F}$ as the canonical representative of ${F}$, so that ${F}$ is now defined everywhere, rather than up to almost everywhere equivalence.

By a classical theorem of Riesz, a function ${F}$ is subharmonic if and only if the distribution ${-\Delta F}$ is a non-negative measure, where ${\Delta := \frac{\partial^2}{\partial \sigma^2} + \frac{\partial^2}{\partial t^2}}$ is the Laplacian in the ${\sigma,t}$ coordinates. Jensen’s formula (or Greens’ theorem), when interpreted distributionally, tells us that

$\displaystyle -\Delta \log |\zeta| = \frac{1}{2\pi} \sum_\rho \delta_\rho$

away from the real axis, where ${\rho}$ ranges over the non-trivial zeroes of ${\zeta}$. Thus, if ${F}$ is a normalised limit profile for ${\log |\zeta|}$ that is the distributional limit of ${F_{T_n}}$, then we have

$\displaystyle -\Delta F = \nu$

where ${\nu}$ is a non-negative measure which is the limit in the vague topology of the measures

$\displaystyle \nu_{T_n} := \frac{1}{2\pi \log T_n} \sum_\rho \delta_{\rho - T_n}.$

Thus ${\nu}$ is a normalised limit profile of the zeroes of the Riemann zeta function.

Using this machinery, we can recover many classical theorems about the Riemann zeta function by “soft” arguments that do not require extensive calculation. Here are some examples:

Theorem 1 The Riemann hypothesis implies the Lindelöf hypothesis.

Proof: It suffices to show that any limiting profile ${F}$ (arising as the limit of some ${F_{T_n}}$) vanishes on the critical line ${\{1/2+it: t \in {\bf R}\}}$. But if the Riemann hypothesis holds, then the measures ${\nu_{T_n}}$ are supported on the critical line ${\{1/2+it: t \in {\bf R}\}}$, so the normalised limit profile ${\nu}$ is also supported on this line. This implies that ${F}$ is harmonic outside of the critical line. By (ii) and unique continuation for harmonic functions, this implies that ${F}$ vanishes on the half-space ${\{ \sigma+it: \sigma \geq \frac{1}{2} \}}$ (and equals ${\frac{1}{2}-\sigma}$ on the complementary half-space, by (iii)), giving the claim. $\Box$

In fact, we have the following sharper statement:

Theorem 2 (Backlund) The Lindelöf hypothesis is equivalent to the assertion that for any fixed ${\sigma_0 > \frac{1}{2}}$, the number of zeroes in the region ${\{ \sigma+it: \sigma > \sigma_0, T \leq t \leq T+1 \}}$ is ${o(\log T)}$ as ${T \rightarrow \infty}$.

Proof: If the latter claim holds, then for any ${T_n \rightarrow \infty}$, the measures ${\nu_{T_n}}$ assign a mass of ${o(1)}$ to any region of the form ${\{ \sigma+it: \sigma > \sigma_0; t_0 \leq t \leq t_0+1 \}}$ as ${n \rightarrow \infty}$ for any fixed ${\sigma_0>\frac{1}{2}}$ and ${t_0 \in {\bf R}}$. Thus the normalised limiting profile measure ${\nu}$ is supported on the critical line, and we can repeat the previous argument.

Conversely, suppose the claim fails, then we can find a sequence ${T_n}$ and ${\sigma_0>0}$ such that ${\nu_{T_n}}$ assigns a mass of ${\gg 1}$ to the region ${\{ \sigma+it: \sigma > \sigma_0; 0\leq t \leq 1 \}}$. Extracting a normalised limiting profile, we conclude that the normalised limiting profile measure ${\nu}$ is non-trivial somewhere to the right of the critical line, so the associated subharmonic function ${F}$ is not harmonic everywhere to the right of the critical line. From the maximum principle and (ii) this implies that ${F}$ has to be positive somewhere on the critical line, but this contradicts the Lindelöf hypothesis. (One has to take a bit of care in the last step since ${F_{T_n}}$ only converges to ${F}$ in the sense of distributions, but it turns out that the subharmonicity of all the functions involved gives enough regularity to justify the argument; we omit the details here.) $\Box$

Theorem 3 (Littlewood) Assume the Lindelöf hypothesis. Then for any fixed ${\alpha>0}$, the number of zeroes in the region ${\{ \sigma+it: T \leq t \leq T+\alpha \}}$ is ${(2\pi \alpha+o(1)) \log T}$ as ${T \rightarrow +\infty}$.

Proof: By the previous arguments, the only possible normalised limiting profile for ${\log |\zeta|}$ is ${\max( 0, \frac{1}{2}-\sigma )}$. Taking distributional Laplacians, we see that the only possible normalised limiting profile for the zeroes is Lebesgue measure on the critical line. Thus, ${\nu_T( \{\sigma+it: T \leq t \leq T+\alpha \} )}$ can only converge to ${\alpha}$ as ${T \rightarrow +\infty}$, and the claim follows. $\Box$

Even without the Lindelöf hypothesis, we have the following result:

Theorem 4 (Titchmarsh) For any fixed ${\alpha>0}$, there are ${\gg_\alpha \log T}$ zeroes in the region ${\{ \sigma+it: T \leq t \leq T+\alpha \}}$ for sufficiently large ${T}$.

Among other things, this theorem recovers a classical result of Littlewood that the gaps between the imaginary parts of the zeroes goes to zero, even without assuming unproven conjectures such as the Riemann or Lindelöf hypotheses.

Proof: Suppose for contradiction that this were not the case, then we can find ${\alpha > 0}$ and a sequence ${T_n \rightarrow \infty}$ such that ${\{ \sigma+it: T_n \leq t \leq T_n+\alpha \}}$ contains ${o(\log T)}$ zeroes. Passing to a subsequence to extract a limit profile, we conclude that the normalised limit profile measure ${\nu}$ assigns no mass to the horizontal strip ${\{ \sigma+it: 0 \leq t \leq\alpha \}}$. Thus the associated subharmonic function ${F}$ is actually harmonic on this strip. But by (ii) and unique continuation this forces ${F}$ to vanish on this strip, contradicting the functional equation (iii). $\Box$

Exercise 5 Use limiting profiles to obtain the matching upper bound of ${O_\alpha(\log T)}$ for the number of zeroes in ${\{ \sigma+it: T \leq t \leq T+\alpha \}}$ for sufficiently large ${T}$.

Remark 6 One can remove the need to take limiting profiles in the above arguments if one can come up with quantitative (or “hard”) substitutes for qualitative (or “soft”) results such as the unique continuation property for harmonic functions. This would also allow one to replace the qualitative decay rates ${o(1)}$ with more quantitative decay rates such as ${1/\log \log T}$ or ${1/\log\log\log T}$. Indeed, the classical proofs of the above theorems come with quantitative bounds that are typically of this form (see e.g. the text of Titchmarsh for details).

Exercise 7 Let ${S(T)}$ denote the quantity ${S(T) := \frac{1}{\pi} \hbox{arg} \zeta(\frac{1}{2}+iT)}$, where the branch of the argument is taken by using a line segment connecting ${\frac{1}{2}+iT}$ to (say) ${2+iT}$, and then to ${2}$. If we have a sequence ${T_n \rightarrow \infty}$ producing normalised limit profiles ${F, \nu}$ for ${\log|\zeta|}$ and the zeroes respectively, show that ${t \mapsto \frac{1}{\log T_n} S(T_n + t)}$ converges in the sense of distributions to the function ${t \mapsto \frac{1}{\pi} \int_{1/2}^1 \frac{\partial F}{\partial t}(\sigma+it)\ d\sigma}$, or equivalently

$\displaystyle t \mapsto \frac{1}{2\pi} \frac{\partial}{\partial t} \int_0^1 F(\sigma+it)\ d\sigma.$

Conclude in particular that if the Lindelöf hypothesis holds, then ${S(T) = o(\log T)}$ as ${T \rightarrow \infty}$.

A little bit more about the normalised limit profiles ${F}$ are known unconditionally, beyond (i)-(iv). For instance, from Exercise 3 of Notes 5 we have ${\zeta(1/2 + it ) = O( t^{1/6+o(1)} )}$ as ${t \rightarrow +\infty}$, which implies that any normalised limit profile ${F}$ for ${\log|\zeta|}$ is bounded by ${1/6}$ on the critical line, beating the bound of ${1/4}$ coming from convexity and (ii), (iii), and then convexity can be used to further bound ${F}$ away from the critical line also. Some further small improvements of this type are known (coming from various methods for estimating exponential sums), though they fall well short of determining ${F}$ completely at our current level of understanding. Of course, given that we believe the Riemann hypothesis (and hence the Lindelöf hypothesis) to be true, the only actual limit profile that should exist is ${\max(0,\frac{1}{2}-\sigma)}$ (in fact this assertion is equivalent to the Lindelöf hypothesis, by the arguments above).

Better control on limiting profiles is available if we do not insist on controlling ${\zeta}$ for all values of the height parameter ${T}$, but only for most such values, thanks to the existence of several mean value theorems for the zeta function, as discussed in Notes 6; we discuss this below the fold.

In analytic number theory, it is a well-known phenomenon that for many arithmetic functions ${f: {\bf N} \rightarrow {\bf C}}$ of interest in number theory, it is significantly easier to estimate logarithmic sums such as

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

than it is to estimate summatory functions such as

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

(Here we are normalising ${f}$ to be roughly constant in size, e.g. ${f(n) = O( n^{o(1)} )}$ as ${n \rightarrow \infty}$.) For instance, when ${f}$ is the von Mangoldt function ${\Lambda}$, the logarithmic sums ${\sum_{n \leq x} \frac{\Lambda(n)}{n}}$ can be adequately estimated by Mertens’ theorem, which can be easily proven by elementary means (see Notes 1); but a satisfactory estimate on the summatory function ${\sum_{n \leq x} \Lambda(n)}$ requires the prime number theorem, which is substantially harder to prove (see Notes 2). (From a complex-analytic or Fourier-analytic viewpoint, the problem is that the logarithmic sums ${\sum_{n \leq x} \frac{f(n)}{n}}$ can usually be controlled just from knowledge of the Dirichlet series ${\sum_n \frac{f(n)}{n^s}}$ for ${s}$ near ${1}$; but the summatory functions require control of the Dirichlet series ${\sum_n \frac{f(n)}{n^s}}$ for ${s}$ on or near a large portion of the line ${\{ 1+it: t \in {\bf R} \}}$. See Notes 2 for further discussion.)

Viewed conversely, whenever one has a difficult estimate on a summatory function such as ${\sum_{n \leq x} f(n)}$, one can look to see if there is a “cheaper” version of that estimate that only controls the logarithmic sums ${\sum_{n \leq x} \frac{f(n)}{n}}$, which is easier to prove than the original, more “expensive” estimate. In this post, we shall do this for two theorems, a classical theorem of Halasz on mean values of multiplicative functions on long intervals, and a much more recent result of Matomaki and Radziwiłł on mean values of multiplicative functions in short intervals. The two are related; the former theorem is an ingredient in the latter (though in the special case of the Matomaki-Radziwiłł theorem considered here, we will not need Halasz’s theorem directly, instead using a key tool in the proof of that theorem).

We begin with Halasz’s theorem. Here is a version of this theorem, due to Montgomery and to Tenenbaum:

Theorem 1 (Halasz-Montgomery-Tenenbaum) Let ${f: {\bf N} \rightarrow {\bf C}}$ be a multiplicative function with ${|f(n)| \leq 1}$ for all ${n}$. Let ${x \geq 3}$ and ${T \geq 1}$, and set

$\displaystyle M := \min_{|t| \leq T} \sum_{p \leq x} \frac{1 - \hbox{Re}( f(p) p^{-it} )}{p}.$

Then one has

$\displaystyle \frac{1}{x} \sum_{n \leq x} f(n) \ll (1+M) e^{-M} + \frac{1}{\sqrt{T}}.$

Informally, this theorem asserts that ${\sum_{n \leq x} f(n)}$ is small compared with ${x}$, unless ${f}$ “pretends” to be like the character ${p \mapsto p^{it}}$ on primes for some small ${y}$. (This is the starting point of the “pretentious” approach of Granville and Soundararajan to analytic number theory, as developed for instance here.) We now give a “cheap” version of this theorem which is significantly weaker (both because it settles for controlling logarithmic sums rather than summatory functions, it requires ${f}$ to be completely multiplicative instead of multiplicative, it requires a strong bound on the analogue of the quantity ${M}$, and because it only gives qualitative decay rather than quantitative estimates), but easier to prove:

Theorem 2 (Cheap Halasz) Let ${x}$ be an asymptotic parameter goingto infinity. Let ${f: {\bf N} \rightarrow {\bf C}}$ be a completely multiplicative function (possibly depending on ${x}$) such that ${|f(n)| \leq 1}$ for all ${n}$, such that

$\displaystyle \sum_{p \leq x} \frac{1 - \hbox{Re}( f(p) )}{p} \gg \log\log x. \ \ \ \ \ (1)$

Then

$\displaystyle \frac{1}{\log x} \sum_{n \leq x} \frac{f(n)}{n} = o(1). \ \ \ \ \ (2)$

Note that now that we are content with estimating exponential sums, we no longer need to preclude the possibility that ${f(p)}$ pretends to be like ${p^{it}}$; see Exercise 11 of Notes 1 for a related observation.

To prove this theorem, we first need a special case of the Turan-Kubilius inequality.

Lemma 3 (Turan-Kubilius) Let ${x}$ be a parameter going to infinity, and let ${1 < P < x}$ be a quantity depending on ${x}$ such that ${P = x^{o(1)}}$ and ${P \rightarrow \infty}$ as ${x \rightarrow \infty}$. Then

$\displaystyle \sum_{n \leq x} \frac{ | \frac{1}{\log \log P} \sum_{p \leq P: p|n} 1 - 1 |}{n} = o( \log x ).$

Informally, this lemma is asserting that

$\displaystyle \sum_{p \leq P: p|n} 1 \approx \log \log P$

for most large numbers ${n}$. Another way of writing this heuristically is in terms of Dirichlet convolutions:

$\displaystyle 1 \approx 1 * \frac{1}{\log\log P} 1_{{\mathcal P} \cap [1,P]}.$

This type of estimate was previously discussed as a tool to establish a criterion of Katai and Bourgain-Sarnak-Ziegler for Möbius orthogonality estimates in this previous blog post. See also Section 5 of Notes 1 for some similar computations.

Proof: By Cauchy-Schwarz it suffices to show that

$\displaystyle \sum_{n \leq x} \frac{ | \frac{1}{\log \log P} \sum_{p \leq P: p|n} 1 - 1 |^2}{n} = o( \log x ).$

Expanding out the square, it suffices to show that

$\displaystyle \sum_{n \leq x} \frac{ (\frac{1}{\log \log P} \sum_{p \leq P: p|n} 1)^j}{n} = \log x + o( \log x )$

for ${j=0,1,2}$.

We just show the ${j=2}$ case, as the ${j=0,1}$ cases are similar (and easier). We rearrange the left-hand side as

$\displaystyle \frac{1}{(\log\log P)^2} \sum_{p_1, p_2 \leq P} \sum_{n \leq x: p_1,p_2|n} \frac{1}{n}.$

We can estimate the inner sum as ${(1+o(1)) \frac{1}{[p_1,p_2]} \log x}$. But a routine application of Mertens’ theorem (handling the diagonal case when ${p_1=p_2}$ separately) shows that

$\displaystyle \sum_{p_1, p_2 \leq P} \frac{1}{[p_1,p_2]} = (1+o(1)) (\log\log P)^2$

and the claim follows. $\Box$

Remark 4 As an alternative to the Turan-Kubilius inequality, one can use the Ramaré identity

$\displaystyle \sum_{p \leq P: p|n} \frac{1}{\# \{ p' \leq P: p'|n\} + 1} - 1 = 1_{(p,n)=1 \hbox{ for all } p \leq P}$

(see e.g. Section 17.3 of Friedlander-Iwaniec). This identity turns out to give superior quantitative results than the Turan-Kubilius inequality in applications; see the paper of Matomaki and Radziwiłł for an instance of this.

We now prove Theorem 2. Let ${Q}$ denote the left-hand side of (2); by the triangle inequality we have ${Q=O(1)}$. By Lemma 3 (for some ${P = x^{o(1)}}$ to be chosen later) and the triangle inequality we have

$\displaystyle \sum_{n \leq x} \frac{\frac{1}{\log \log P} \sum_{p \leq P: p|n} f(n)}{n} = Q \log x + o( \log x ).$

We rearrange the left-hand side as

$\displaystyle \frac{1}{\log\log P} \sum_{p \leq P} \frac{f(p)}{p} \sum_{m \leq x/p} \frac{f(m)}{m}.$

We now replace the constraint ${m \leq x/p}$ by ${m \leq x}$. The error incurred in doing so is

$\displaystyle O( \frac{1}{\log\log P} \sum_{p \leq P} \frac{1}{p} \sum_{x/P \leq m \leq x} \frac{1}{m} )$

which by Mertens’ theorem is ${O(\log P) = o( \log x )}$. Thus we have

$\displaystyle \frac{1}{\log\log P} \sum_{p \leq P} \frac{f(p)}{p} \sum_{m \leq x} \frac{f(m)}{m} = Q \log x + o( \log x ).$

But by definition of ${Q}$, we have ${\sum_{m \leq x} \frac{f(m)}{m} = Q \log x}$, thus

$\displaystyle [1 - \frac{1}{\log\log P} \sum_{p \leq P} \frac{f(p)}{p}] Q = o(1). \ \ \ \ \ (3)$

From Mertens’ theorem, the expression in brackets can be rewritten as

$\displaystyle \frac{1}{\log\log P} \sum_{p \leq P} \frac{1 - f(p)}{p} + o(1)$

and so the real part of this expression is

$\displaystyle \frac{1}{\log\log P} \sum_{p \leq P} \frac{1 - \hbox{Re} f(p)}{p} + o(1).$

By (1), Mertens’ theorem and the hypothesis on ${f}$ we have

$\displaystyle \sum_{p \leq x^\varepsilon} \frac{(1 - \hbox{Re} f(p)) \log p}{p} \gg \log\log x^\varepsilon - O_\varepsilon(1)$

for any ${\varepsilon > 0}$. This implies that we can find ${P = x^{o(1)}}$ going to infinity such that

$\displaystyle \sum_{p \leq P} \frac{(1 - \hbox{Re} f(p)) \log p}{p} \gg (1-o(1))\log\log P$

and thus the expression in brackets has real part ${\gg 1-o(1)}$. The claim follows.

The Turan-Kubilius argument is certainly not the most efficient way to estimate sums such as ${\frac{1}{n} \sum_{n \leq x} f(n)}$. In the exercise below we give a significantly more accurate estimate that works when ${f}$ is non-negative.

Exercise 5 (Granville-Koukoulopoulos-Matomaki)

• (i) If ${g}$ is a completely multiplicative function with ${g(p) \in \{0,1\}}$ for all primes ${p}$, show that

$\displaystyle (e^{-\gamma}-o(1)) \prod_{p \leq x} (1 - \frac{g(p)}{p})^{-1} \leq \sum_{n \leq x} \frac{g(n)}{n} \leq \prod_{p \leq x} (1 - \frac{g(p)}{p})^{-1}.$

as ${x \rightarrow \infty}$. (Hint: for the upper bound, expand out the Euler product. For the lower bound, show that ${\sum_{n \leq x} \frac{g(n)}{n} \times \sum_{n \leq x} \frac{h(n)}{n} \ge \sum_{n \leq x} \frac{1}{n}}$, where ${h}$ is the completely multiplicative function with ${h(p) = 1-g(p)}$ for all primes ${p}$.)

• (ii) If ${g}$ is multiplicative and takes values in ${[0,1]}$, show that

$\displaystyle \sum_{n \leq x} \frac{g(n)}{n} \asymp \prod_{p \leq x} (1 - \frac{g(p)}{p})^{-1}$

$\displaystyle \asymp \exp( \sum_{p \leq x} \frac{g(p)}{p} )$

for all ${x \geq 1}$.

Now we turn to a very recent result of Matomaki and Radziwiłł on mean values of multiplicative functions in short intervals. For sake of illustration we specialise their results to the simpler case of the Liouville function ${\lambda}$, although their arguments actually work (with some additional effort) for arbitrary multiplicative functions of magnitude at most ${1}$ that are real-valued (or more generally, stay far from complex characters ${p \mapsto p^{it}}$). Furthermore, we give a qualitative form of their estimates rather than a quantitative one:

Theorem 6 (Matomaki-Radziwiłł, special case) Let ${X}$ be a parameter going to infinity, and let ${2 \leq h \leq X}$ be a quantity going to infinity as ${X \rightarrow \infty}$. Then for all but ${o(X)}$ of the integers ${x \in [X,2X]}$, one has

$\displaystyle \sum_{x \leq n \leq x+h} \lambda(n) = o( h ).$

Equivalently, one has

$\displaystyle \sum_{X \leq x \leq 2X} |\sum_{x \leq n \leq x+h} \lambda(n)|^2 = o( h^2 X ). \ \ \ \ \ (4)$

A simple sieving argument (see Exercise 18 of Supplement 4) shows that one can replace ${\lambda}$ by the Möbius function ${\mu}$ and obtain the same conclusion. See this recent note of Matomaki and Radziwiłł for a simple proof of their (quantitative) main theorem in this special case.

Of course, (4) improves upon the trivial bound of ${O( h^2 X )}$. Prior to this paper, such estimates were only known (using arguments similar to those in Section 3 of Notes 6) for ${h \geq X^{1/6+\varepsilon}}$ unconditionally, or for ${h \geq \log^A X}$ for some sufficiently large ${A}$ if one assumed the Riemann hypothesis. This theorem also represents some progress towards Chowla’s conjecture (discussed in Supplement 4) that

$\displaystyle \sum_{n \leq x} \lambda(n+h_1) \dots \lambda(n+h_k) = o( x )$

as ${x \rightarrow \infty}$ for any fixed distinct ${h_1,\dots,h_k}$; indeed, it implies that this conjecture holds if one performs a small amount of averaging in the ${h_1,\dots,h_k}$.

Below the fold, we give a “cheap” version of the Matomaki-Radziwiłł argument. More precisely, we establish

Theorem 7 (Cheap Matomaki-Radziwiłł) Let ${X}$ be a parameter going to infinity, and let ${1 \leq T \leq X}$. Then

$\displaystyle \int_X^{X^A} \left|\sum_{x \leq n \leq e^{1/T} x} \frac{\lambda(n)}{n}\right|^2\frac{dx}{x} = o\left( \frac{\log X}{T^2} \right), \ \ \ \ \ (5)$

for any fixed ${A>1}$.

Note that (5) improves upon the trivial bound of ${O( \frac{\log X}{T^2} )}$. Again, one can replace ${\lambda}$ with ${\mu}$ if desired. Due to the cheapness of Theorem 7, the proof will require few ingredients; the deepest input is the improved zero-free region for the Riemann zeta function due to Vinogradov and Korobov. Other than that, the main tools are the Turan-Kubilius result established above, and some Fourier (or complex) analysis.

In the previous set of notes, we saw how zero-density theorems for the Riemann zeta function, when combined with the zero-free region of Vinogradov and Korobov, could be used to obtain prime number theorems in short intervals. It turns out that a more sophisticated version of this type of argument also works to obtain prime number theorems in arithmetic progressions, in particular establishing the celebrated theorem of Linnik:

Theorem 1 (Linnik’s theorem) Let ${a\ (q)}$ be a primitive residue class. Then ${a\ (q)}$ contains a prime ${p}$ with ${p \ll q^{O(1)}}$.

In fact it is known that one can find a prime ${p}$ with ${p \ll q^{5}}$, a result of Xylouris. For sake of comparison, recall from Exercise 65 of Notes 2 that the Siegel-Walfisz theorem gives this theorem with a bound of ${p \ll \exp( q^{o(1)} )}$, and from Exercise 48 of Notes 2 one can obtain a bound of the form ${p \ll \phi(q)^2 \log^2 q}$ if one assumes the generalised Riemann hypothesis. The probabilistic random models from Supplement 4 suggest that one should in fact be able to take ${p \ll q^{1+o(1)}}$.

We will not aim to obtain the optimal exponents for Linnik’s theorem here, and follow the treatment in Chapter 18 of Iwaniec and Kowalski. We will in fact establish the following more quantitative result (a special case of a more powerful theorem of Gallagher), which splits into two cases, depending on whether there is an exceptional zero or not:

Theorem 2 (Quantitative Linnik theorem) Let ${a\ (q)}$ be a primitive residue class for some ${q \geq 2}$. For any ${x > 1}$, let ${\psi(x;q,a)}$ denote the quantity

$\displaystyle \psi(x;q,a) := \sum_{n \leq x: n=a\ (q)} \Lambda(n).$

Assume that ${x \geq q^C}$ for some sufficiently large ${C}$.

• (i) (No exceptional zero) If all the real zeroes ${\beta}$ of ${L}$-functions ${L(\cdot,\chi)}$ of real characters ${\chi}$ of modulus ${q}$ are such that ${1-\beta \gg \frac{1}{\log q}}$, then

$\displaystyle \psi(x;q,a) = \frac{x}{\phi(q)} ( 1 + O( \exp( - c \frac{\log x}{\log q} ) ) + O( \frac{\log^2 q}{q} ) )$

for all ${x \geq 1}$ and some absolute constant ${c>0}$.

• (ii) (Exceptional zero) If there is a zero ${\beta}$ of an ${L}$-function ${L(\cdot,\chi_1)}$ of a real character ${\chi_1}$ of modulus ${q}$ with ${\beta = 1 - \frac{\varepsilon}{\log q}}$ for some sufficiently small ${\varepsilon>0}$, then

$\displaystyle \psi(x;q,a) = \frac{x}{\phi(q)} ( 1 - \chi_1(a) \frac{x^{\beta-1}}{\beta} \ \ \ \ \ (1)$

$\displaystyle + O( \exp( - c \frac{\log x}{\log q} \log \frac{1}{\varepsilon} ) )$

$\displaystyle + O( \frac{\log^2 q}{q} ) )$

for all ${x \geq 1}$ and some absolute constant ${c>0}$.

The implied constants here are effective.

Note from the Landau-Page theorem (Exercise 54 from Notes 2) that at most one exceptional zero exists (if ${\varepsilon}$ is small enough). A key point here is that the error term ${O( \exp( - c \frac{\log x}{\log q} \log \frac{1}{\varepsilon} ) )}$ in the exceptional zero case is an improvement over the error term when no exceptional zero is present; this compensates for the potential reduction in the main term coming from the ${\chi_1(a) \frac{x^{\beta-1}}{\beta}}$ term. The splitting into cases depending on whether an exceptional zero exists or not turns out to be an essential technique in many advanced results in analytic number theory (though presumably such a splitting will one day become unnecessary, once the possibility of exceptional zeroes are finally eliminated for good).

Exercise 3 Assuming Theorem 2, and assuming ${x \geq q^C}$ for some sufficiently large absolute constant ${C}$, establish the lower bound

$\displaystyle \psi(x;a,q) \gg \frac{x}{\phi(q)}$

when there is no exceptional zero, and

$\displaystyle \psi(x;a,q) \gg \varepsilon \frac{x}{\phi(q)}$

when there is an exceptional zero ${\beta = 1 - \frac{\varepsilon}{\log q}}$. Conclude that Theorem 2 implies Theorem 1, regardless of whether an exceptional zero exists or not.

Remark 4 The Brun-Titchmarsh theorem (Exercise 33 from Notes 4), in the sharp form of Montgomery and Vaughan, gives that

$\displaystyle \pi(x; q, a) \leq 2 \frac{x}{\phi(q) \log (x/q)}$

for any primitive residue class ${a\ (q)}$ and any ${x \geq q}$. This is (barely) consistent with the estimate (1). Any lowering of the coefficient ${2}$ in the Brun-Titchmarsh inequality (with reasonable error terms), in the regime when ${x}$ is a large power of ${q}$, would then lead to at least some elimination of the exceptional zero case. However, this has not led to any progress on the Landau-Siegel zero problem (and may well be just a reformulation of that problem). (When ${x}$ is a relatively small power of ${q}$, some improvements to Brun-Titchmarsh are possible that are not in contradiction with the presence of an exceptional zero; see this paper of Maynard for more discussion.)

Theorem 2 is deduced in turn from facts about the distribution of zeroes of ${L}$-functions. We first need a version of the truncated explicit formula that does not lose unnecessary logarithms:

Exercise 5 (Log-free truncated explicit formula) With the hypotheses as above, show that

$\displaystyle \sum_{n \leq x} \Lambda(n) \chi(n) = - \sum_{\hbox{Re}(\rho) > 3/4; |\hbox{Im}(\rho)| \leq q^2; L(\rho,\chi)=0} \frac{x^\rho}{\rho} + O( \frac{x}{q^2} \log^2 q)$

for any non-principal character ${\chi}$ of modulus ${q}$, where we assume ${x \geq q^C}$ for some large ${C}$; for the principal character establish the same formula with an additional term of ${x}$ on the right-hand side. (Hint: this is almost immediate from Exercise 45(iv) and Theorem 21 ofNotes 2) with (say) ${T := q^2}$, except that there is a factor of ${\log^2 x}$ in the error term instead of ${\log^2 q}$ when ${x}$ is extremely large compared to ${q}$. However, a closer inspection of the proof (particularly with regards to the truncated Perron formula in Proposition 12 of Notes 2) shows that the ${\log^2 x}$ factor can be replaced fairly easily by ${\log x \log q}$. To get rid of the final factor of ${\log x}$, note that the proof of Proposition 12 used the rather crude bound ${\Lambda(n) = O(\log n)}$. If one replaces this crude bound by more sophisticated tools such as the Brun-Titchmarsh inequality, one will be able to remove the factor of ${\log x}$.

Using the Fourier inversion formula

$\displaystyle 1_{n = a\ (q)} = \frac{1}{\phi(q)} \sum_{\chi\ (q)} \overline{\chi(a)} \chi(n)$

(see Theorem 69 of Notes 1), we thus have

$\displaystyle \psi(x;a,q) = \frac{x}{\phi(q)} ( 1 - \sum_{\chi\ (q)} \overline{\chi(a)} \sum_{\hbox{Re}(\rho) > 3/4; |\hbox{Im}(\rho)| \leq q^2; L(\rho,\chi)=0} \frac{x^{\rho-1}}{\rho}$

$\displaystyle + O( \frac{\log^2 q}{q} ) )$

and so it suffices by the triangle inequality (bounding ${1/\rho}$ very crudely by ${O(1)}$, as the contribution of the low-lying zeroes already turns out to be quite dominant) to show that

$\displaystyle \sum_{\chi\ (q)} \sum_{\sigma > 3/4; |t| \leq q^2; L(\sigma+it,\chi)=0} x^{\sigma-1} \ll \exp( - c \frac{\log x}{\log q} ) \ \ \ \ \ (2)$

when no exceptional zero is present, and

$\displaystyle \sum_{\chi\ (q)} \sum_{\sigma > 3/4; |t| \leq q^2; L(\sigma+it,\chi)=0; \sigma+it \neq \beta} x^{\sigma-1} \ll \exp( - c \frac{\log x}{\log q} \log \frac{1}{\varepsilon} ) \ \ \ \ \ (3)$

when an exceptional zero is present.

To handle the former case (2), one uses two facts about zeroes. The first is the classical zero-free region (Proposition 51 from Notes 2), which we reproduce in our context here:

Proposition 6 (Classical zero-free region) Let ${q, T \geq 2}$. Apart from a potential exceptional zero ${\beta}$, all zeroes ${\sigma+it}$ of ${L}$-functions ${L(\cdot,\chi)}$ with ${\chi}$ of modulus ${q}$ and ${|t| \leq T}$ are such that

$\displaystyle \sigma \leq 1 - \frac{c}{\log qT}$

for some absolute constant ${c>0}$.

Using this zero-free region, we have

$\displaystyle x^{\sigma-1} \ll \log x \int_{1/2}^{1-c/\log q} 1_{\alpha < \sigma} x^{\alpha-1}\ d\alpha$

whenever ${\sigma}$ contributes to the sum in (2), and so the left-hand side of (2) is bounded by

$\displaystyle \ll \log x \int_{1/2}^{1 - c/\log q} N( \alpha, q, q^2 ) x^{\alpha-1}\ d\alpha$

where we recall that ${N(\alpha,q,T)}$ is the number of zeroes ${\sigma+it}$ of any ${L}$-function of a character ${\chi}$ of modulus ${q}$ with ${\sigma \geq \alpha}$ and ${0 \leq t \leq T}$ (here we use conjugation symmetry to make ${t}$ non-negative, accepting a multiplicative factor of two).

In Exercise 25 of Notes 6, the grand density estimate

$\displaystyle N(\alpha,q,T) \ll (qT)^{4(1-\alpha)} \log^{O(1)}(qT) \ \ \ \ \ (4)$

is proven. If one inserts this bound into the above expression, one obtains a bound for (2) which is of the form

$\displaystyle \ll (\log^{O(1)} q) \exp( - c \frac{\log x}{\log q} ).$

Unfortunately this is off from what we need by a factor of ${\log^{O(1)} q}$ (and would lead to a weak form of Linnik’s theorem in which ${p}$ was bounded by ${O( \exp( \log^{O(1)} q ) )}$ rather than by ${q^{O(1)}}$). In the analogous problem for prime number theorems in short intervals, we could use the Vinogradov-Korobov zero-free region to compensate for this loss, but that region does not help here for the contribution of the low-lying zeroes with ${t = O(1)}$, which as mentioned before give the dominant contribution. Fortunately, it is possible to remove this logarithmic loss from the zero-density side of things:

Theorem 7 (Log-free grand density estimate) For any ${q, T > 1}$ and ${1/2 \leq \alpha \leq 1}$, one has

$\displaystyle N(\alpha,q,T) \ll (qT)^{O(1-\alpha)}.$

The implied constants are effective.

We prove this estimate below the fold. The proof follows the methods of the previous section, but one inserts various sieve weights to restrict sums over natural numbers to essentially become sums over “almost primes”, as this turns out to remove the logarithmic losses. (More generally, the trick of restricting to almost primes by inserting suitable sieve weights is quite useful for avoiding any unnecessary losses of logarithmic factors in analytic number theory estimates.)

Exercise 8 Use Theorem 7 to complete the proof of (2).

Now we turn to the case when there is an exceptional zero (3). The argument used to prove (2) applies here also, but does not gain the factor of ${\log \frac{1}{\varepsilon}}$ in the exponent. To achieve this, we need an additional tool, a version of the Deuring-Heilbronn repulsion phenomenon due to Linnik:

Theorem 9 (Deuring-Heilbronn repulsion phenomenon) Suppose ${q \geq 2}$ is such that there is an exceptional zero ${\beta = 1 - \frac{\varepsilon}{\log q}}$ with ${\varepsilon}$ small. Then all other zeroes ${\sigma+it}$ of ${L}$-functions of modulus ${q}$ are such that

$\displaystyle \sigma \leq 1 - c \frac{\log \frac{1}{\varepsilon}}{\log(q(2+|t|))}.$

In other words, the exceptional zero enlarges the classical zero-free region by a factor of ${\log \frac{1}{\varepsilon}}$. The implied constants are effective.

Exercise 10 Use Theorem 7 and Theorem 9 to complete the proof of (3), and thus Linnik’s theorem.

Exercise 11 Use Theorem 9 to give an alternate proof of (Tatuzawa’s version of) Siegel’s theorem (Theorem 62 of Notes 2). (Hint: if two characters have different moduli, then they can be made to have the same modulus by multiplying by suitable principal characters.)

Theorem 9 is proven by similar methods to that of Theorem 7, the basic idea being to insert a further weight of ${1 * \chi_1}$ (in addition to the sieve weights), the point being that the exceptional zero causes this weight to be quite small on the average. There is a strengthening of Theorem 9 due to Bombieri that is along the lines of Theorem 7, obtaining the improvement

$\displaystyle N'(\alpha,q,T) \ll \varepsilon (1 + \frac{\log T}{\log q}) (qT)^{O(1-\alpha)} \ \ \ \ \ (5)$

with effective implied constants for any ${1/2 \leq \alpha \leq 1}$ and ${T \geq 1}$ in the presence of an exceptional zero, where the prime in ${N'(\alpha,q,T)}$ means that the exceptional zero ${\beta}$ is omitted (thus ${N'(\alpha,q,T) = N(\alpha,q,T)-1}$ if ${\alpha \leq \beta}$). Note that the upper bound on ${N'(\alpha,q,T)}$ falls below one when ${\alpha > 1 - c \frac{\log \frac{1}{\varepsilon}}{\log(qT)}}$ for a sufficiently small ${c>0}$, thus recovering Theorem 9. Bombieri’s theorem can be established by the methods in this set of notes, and will be given as an exercise to the reader.

Remark 12 There are a number of alternate ways to derive the results in this set of notes, for instance using the Turan power sums method which is based on studying derivatives such as

$\displaystyle \frac{L'}{L}(s,\chi)^{(k)} = (-1)^k \sum_n \frac{\Lambda(n) \chi(n) \log^k n}{n^s}$

$\displaystyle \approx (-1)^{k+1} k! \sum_\rho \frac{1}{(s-\rho)^{k+1}}$

for ${\hbox{Re}(s)>1}$ and large ${k}$, and performing various sorts of averaging in ${k}$ to attenuate the contribution of many of the zeroes ${\rho}$. We will not develop this method here, but see for instance Chapter 9 of Montgomery’s book. See the text of Friedlander and Iwaniec for yet another approach based primarily on sieve-theoretic ideas.

Remark 13 When one optimises all the exponents, it turns out that the exponent in Linnik’s theorem is extremely good in the presence of an exceptional zero – indeed Friedlander and Iwaniec showed can even get a bound of the form ${p \ll q^{2-c}}$ for some ${c>0}$, which is even stronger than one can obtain from GRH! There are other places in which exceptional zeroes can be used to obtain results stronger than what one can obtain even on the Riemann hypothesis; for instance, Heath-Brown used the hypothesis of an infinite sequence of Siegel zeroes to obtain the twin prime conejcture.

In the previous set of notes, we studied upper bounds on sums such as ${|\sum_{N \leq n \leq N+M} n^{-it}|}$ for ${1 \leq M \leq N}$ that were valid for all ${t}$ in a given range, such as ${[T,2T]}$; this led in turn to upper bounds on the Riemann zeta ${\zeta(\sigma+it)}$ for ${t}$ in the same range, and for various choices of ${\sigma}$. While some improvement over the trivial bound of ${O(N)}$ was obtained by these methods, we did not get close to the conjectural bound of ${O( N^{1/2+o(1)})}$ that one expects from pseudorandomness heuristics (assuming that ${T}$ is not too large compared with ${N}$, e.g. ${T = O(N^{O(1)})}$.

However, it turns out that one can get much better bounds if one settles for estimating sums such as ${|\sum_{N \leq n \leq N+M} n^{-it}|}$, or more generally finite Dirichlet series (also known as Dirichlet polynomials) such as ${|\sum_n a_n n^{-it}|}$, for most values of ${t}$ in a given range such as ${[T,2T]}$. Equivalently, we will be able to get some control on the large values of such Dirichlet polynomials, in the sense that we can control the set of ${t}$ for which ${|\sum_n a_n n^{-it}|}$ exceeds a certain threshold, even if we cannot show that this set is empty. These large value theorems are often closely tied with estimates for mean values such as ${\frac{1}{T}\int_T^{2T} |\sum_n a_n n^{-it}|^{2k}\ dt}$ of a Dirichlet series; these latter estimates are thus known as mean value theorems for Dirichlet series. Our approach to these theorems will follow the same sort of methods used in Notes 3, in particular relying on the generalised Bessel inequality from those notes.

Our main application of the large value theorems for Dirichlet polynomials will be to control the number of zeroes of the Riemann zeta function ${\zeta(s)}$ (or the Dirichlet ${L}$-functions ${L(s,\chi)}$) in various rectangles of the form ${\{ \sigma+it: \sigma \geq \alpha, |t| \leq T \}}$ for various ${T > 1}$ and ${1/2 < \alpha < 1}$. These rectangles will be larger than the zero-free regions for which we can exclude zeroes completely, but we will often be able to limit the number of zeroes in such rectangles to be quite small. For instance, we will be able to show the following weak form of the Riemann hypothesis: as ${T \rightarrow \infty}$, a proportion ${1-o(1)}$ of zeroes of the Riemann zeta function in the critical strip with ${|\hbox{Im}(s)| \leq T}$ will have real part ${1/2+o(1)}$. Related to this, the number of zeroes with ${|\hbox{Im}(s)| \leq T}$ and ${|\hbox{Re}(s)| \geq \alpha}$ can be shown to be bounded by ${O( T^{O(1-\alpha)+o(1)} )}$ as ${T \rightarrow \infty}$ for any ${1/2 < \alpha < 1}$.

In the next set of notes we will use refined versions of these theorems to establish Linnik’s theorem on the least prime in an arithmetic progression.

Our presentation here is broadly based on Chapters 9 and 10 in Iwaniec and Kowalski, who give a number of more sophisticated large value theorems than the ones discussed here.

We return to the study of the Riemann zeta function ${\zeta(s)}$, focusing now on the task of upper bounding the size of this function within the critical strip; as seen in Exercise 43 of Notes 2, such upper bounds can lead to zero-free regions for ${\zeta}$, which in turn lead to improved estimates for the error term in the prime number theorem.

In equation (21) of Notes 2 we obtained the somewhat crude estimates

$\displaystyle \zeta(s) = \sum_{n \leq x} \frac{1}{n^s} - \frac{x^{1-s}}{1-s} + O( \frac{|s|}{\sigma} \frac{1}{x^\sigma} ) \ \ \ \ \ (1)$

for any ${x > 0}$ and ${s = \sigma+it}$ with ${\sigma>0}$ and ${s \neq 1}$. Setting ${x=1}$, we obtained the crude estimate

$\displaystyle \zeta(s) = \frac{1}{s-1} + O( \frac{|s|}{\sigma} )$

in this region. In particular, if ${0 < \varepsilon \leq \sigma \ll 1}$ and ${|t| \gg 1}$ then we had ${\zeta(s) = O_\varepsilon( |t| )}$. Using the functional equation and the Hadamard three lines lemma, we can improve this to ${\zeta(s) \ll_\varepsilon |t|^{\frac{1-\sigma}{2}+\varepsilon}}$; see Supplement 3.
Now we seek better upper bounds on ${\zeta}$. We will reduce the problem to that of bounding certain exponential sums, in the spirit of Exercise 34 of Supplement 3:

Proposition 1 Let ${s = \sigma+it}$ with ${0 < \varepsilon \leq \sigma \ll 1}$ and ${|t| \gg 1}$. Then

$\displaystyle \zeta(s) \ll_\varepsilon \log(2+|t|) \sup_{1 \leq M \leq N \ll |t|}$

$\displaystyle N^{1-\sigma} |\frac{1}{N} \sum_{N \leq n < N+M} e( -\frac{t}{2\pi} \log n)|$

where ${e(x) := e^{2\pi i x}}$.

Proof: We fix a smooth function ${\eta: {\bf R} \rightarrow {\bf C}}$ with ${\eta(t)=1}$ for ${t \leq -1}$ and ${\eta(t)=0}$ for ${t \geq 1}$, and allow implied constants to depend on ${\eta}$. Let ${s=\sigma+it}$ with ${\varepsilon \leq \sigma \ll 1}$. From Exercise 34 of Supplement 3, we have

$\displaystyle \zeta(s) = \sum_n \frac{1}{n^s} \eta( \log n - \log C|t| ) + O_\varepsilon( 1 )$

for some sufficiently large absolute constant ${C}$. By dyadic decomposition, we thus have

$\displaystyle \zeta(s) \ll_{\varepsilon} 1 + \log(2+|t|) \sup_{1 \leq N \ll |t|} |\sum_{N \leq n < 2N} \frac{1}{n^s} \eta( \log n - \log C|t| )|.$

We can absorb the first term in the second using the ${N=1}$ case of the supremum. Writing ${\frac{1}{n^s} \eta( \log n - \log|C| t ) = N^{-\sigma} e( - \frac{t}{2\pi} \log n ) F_N(n)}$, where

$\displaystyle F_N(n) := (N/n)^\sigma \eta(\log n - \log C|t| ),$

it thus suffices to show that

$\displaystyle \sum_{N \leq n < 2N} e(-\frac{t}{2\pi} \log N) F_N(n) \ll \sup_{1 \leq M \leq N} |\sum_{N \leq n < N+M} e(-\frac{t}{2\pi} \log n)|$

for each ${N}$. But from the fundamental theorem of calculus, the left-hand side can be written as

$\displaystyle F_N(2N) \sum_{N \leq n < 2N} e(-\frac{t}{2\pi} \log n)$

$\displaystyle - \int_0^{N} (\sum_{N \leq n < N+M} e(-\frac{t}{2\pi} \log n)) F'_N(M)\ dM$

and the claim then follows from the triangle inequality and a routine calculation. $\Box$
We are thus interested in getting good bounds on the sum ${\sum_{N \leq n < N+M} e( -\frac{t}{2\pi} \log n )}$. More generally, we consider normalised exponential sums of the form

$\displaystyle \frac{1}{N} \sum_{n \in I} e( f(n) ) \ \ \ \ \ (2)$

where ${I \subset {\bf R}}$ is an interval of length at most ${N}$ for some ${N \geq 1}$, and ${f: {\bf R} \rightarrow {\bf R}}$ is a smooth function. We will assume smoothness estimates of the form

$\displaystyle |f^{(j)}(x)| = \exp( O(j^2) ) \frac{T}{N^j} \ \ \ \ \ (3)$

for some ${T>0}$, all ${x \in I}$, and all ${j \geq 1}$, where ${f^{(j)}}$ is the ${j}$-fold derivative of ${f}$; in the case ${f(x) := -\frac{t}{2\pi} \log x}$, ${I \subset [N,2N]}$ of interest for the Riemann zeta function, we easily verify that these estimates hold with ${T := |t|}$. (One can consider exponential sums under more general hypotheses than (3), but the hypotheses here are adequate for our needs.) We do not bound the zeroth derivative ${f^{(0)}=f}$ of ${f}$ directly, but it would not be natural to do so in any event, since the magnitude of the sum (2) is unaffected if one adds an arbitrary constant to ${f(n)}$.
The trivial bound for (2) is

$\displaystyle \frac{1}{N} \sum_{n \in I} e(f(n)) \ll 1 \ \ \ \ \ (4)$

and we will seek to obtain significant improvements to this bound. Pseudorandomness heuristics predict a bound of ${O_\varepsilon(N^{-1/2+\varepsilon})}$ for (2) for any ${\varepsilon>0}$ if ${T = O(N^{O(1)})}$; this assertion (a special case of the exponent pair hypothesis) would have many consequences (for instance, inserting it into Proposition 1 soon yields the Lindelöf hypothesis), but is unfortunately quite far from resolution with known methods. However, we can obtain weaker gains of the form ${O(N^{1-c_K})}$ when ${T \ll N^K}$ and ${c_K > 0}$ depends on ${K}$. We present two such results here, which perform well for small and large values of ${K}$ respectively:

Theorem 2 Let ${2 \leq N \ll T}$, let ${I}$ be an interval of length at most ${N}$, and let ${f: I \rightarrow {\bf R}}$ be a smooth function obeying (3) for all ${j \geq 1}$ and ${x \in I}$.

• (i) (van der Corput estimate) For any natural number ${k \geq 2}$, one has

$\displaystyle \frac{1}{N} \sum_{n \in I} e( f(n) ) \ll (\frac{T}{N^k})^{\frac{1}{2^k-2}} \log^{1/2} (2+T). \ \ \ \ \ (5)$

• (ii) (Vinogradov estimate) If ${k}$ is a natural number and ${T \leq N^{k}}$, then

$\displaystyle \frac{1}{N} \sum_{n \in I} e( f(n) ) \ll N^{-c/k^2} \ \ \ \ \ (6)$

for some absolute constant ${c>0}$.

The factor of ${\log^{1/2} (2+T)}$ can be removed by a more careful argument, but we will not need to do so here as we are willing to lose powers of ${\log T}$. The estimate (6) is superior to (5) when ${T \sim N^K}$ for ${K}$ large, since (after optimising in ${k}$) (5) gives a gain of the form ${N^{-c/2^{cK}}}$ over the trivial bound, while (6) gives ${N^{-c/K^2}}$. We have not attempted to obtain completely optimal estimates here, settling for a relatively simple presentation that still gives good bounds on ${\zeta}$, and there are a wide variety of additional exponential sum estimates beyond the ones given here; see Chapter 8 of Iwaniec-Kowalski, or Chapters 3-4 of Montgomery, for further discussion.

We now briefly discuss the strategies of proof of Theorem 2. Both parts of the theorem proceed by treating ${f}$ like a polynomial of degree roughly ${k}$; in the case of (ii), this is done explicitly via Taylor expansion, whereas for (i) it is only at the level of analogy. Both parts of the theorem then try to “linearise” the phase to make it a linear function of the summands (actually in part (ii), it is necessary to introduce an additional variable and make the phase a bilinear function of the summands). The van der Corput estimate achieves this linearisation by squaring the exponential sum about ${k}$ times, which is why the gain is only exponentially small in ${k}$. The Vinogradov estimate achieves linearisation by raising the exponential sum to a significantly smaller power – on the order of ${k^2}$ – by using Hölder’s inequality in combination with the fact that the discrete curve ${\{ (n,n^2,\dots,n^k): n \in \{1,\dots,M\}\}}$ becomes roughly equidistributed in the box ${\{ (a_1,\dots,a_k): a_j = O( M^j ) \}}$ after taking the sumset of about ${k^2}$ copies of this curve. This latter fact has a precise formulation, known as the Vinogradov mean value theorem, and its proof is the most difficult part of the argument, relying on using a “${p}$-adic” version of this equidistribution to reduce the claim at a given scale ${M}$ to a smaller scale ${M/p}$ with ${p \sim M^{1/k}}$, and then proceeding by induction.

One can combine Theorem 2 with Proposition 1 to obtain various bounds on the Riemann zeta function:

Exercise 3 (Subconvexity bound)

• (i) Show that ${\zeta(\frac{1}{2}+it) \ll (1+|t|)^{1/6} \log^{O(1)}(2+|t|)}$ for all ${t \in {\bf R}}$. (Hint: use the ${k=3}$ case of the Van der Corput estimate.)
• (ii) For any ${0 < \sigma < 1}$, show that ${\zeta(\sigma+it) \ll (1+|t|)^{\max( \frac{1-\sigma}{3}, \frac{1}{2} - \frac{2\sigma}{3}) + o(1)}}$ as ${|t| \rightarrow \infty}$ (the decay rate in the ${o(1)}$ is allowed to depend on ${\sigma}$).

Exercise 4 Let ${t}$ be such that ${|t| \geq 100}$, and let ${\sigma \geq 1/2}$.

• (i) (Littlewood bound) Use the van der Corput estimate to show that ${\zeta(\sigma+it) \ll \log^{O(1)} |t|}$ whenever ${\sigma \geq 1 - O( \frac{(\log\log |t|)^2}{\log |t|} ))}$.
• (ii) (Vinogradov-Korobov bound) Use the Vinogradov estimate to show that ${\zeta(\sigma+it) \ll \log^{O(1)} |t|}$ whenever ${\sigma \geq 1 - O( \frac{(\log\log |t|)^{2/3}}{\log^{2/3} |t|} )}$.

As noted in Exercise 43 of Notes 2, the Vinogradov-Korobov bound leads to the zero-free region ${\{ \sigma+it: \sigma > 1 - c \frac{1}{(\log |t|)^{2/3} (\log\log |t|)^{1/3}}; |t| \geq 100 \}}$, which in turn leads to the prime number theorem with error term

$\displaystyle \sum_{n \leq x} \Lambda(n) = x + O\left( x \exp\left( - c \frac{\log^{3/5} x}{(\log\log x)^{1/5}} \right) \right)$

for ${x > 100}$. If one uses the weaker Littlewood bound instead, one obtains the narrower zero-free region

$\displaystyle \{ \sigma+it: \sigma > 1 - c \frac{\log\log|t|}{\log |t|}; |t| \geq 100 \}$

(which is only slightly wider than the classical zero-free region) and an error term

$\displaystyle \sum_{n \leq x} \Lambda(n) = x + O( x \exp( - c \sqrt{\log x \log\log x} ) )$

in the prime number theorem.

Exercise 5 (Vinogradov-Korobov in arithmetic progressions) Let ${\chi}$ be a non-principal character of modulus ${q}$.

• (i) (Vinogradov-Korobov bound) Use the Vinogradov estimate to show that ${L(\sigma+it,\chi) \ll \log^{O(1)}(q|t|)}$ whenever ${|t| \geq 100}$ and

$\displaystyle \sigma \geq 1 - O( \min( \frac{\log\log(q|t|)}{\log q}, \frac{(\log\log(q|t|))^{2/3}}{\log^{2/3} |t|} ) ).$

(Hint: use the Vinogradov estimate and a change of variables to control ${\sum_{n \in I: n = a\ (q)} \exp( -it \log n)}$ for various intervals ${I}$ of length at most ${N}$ and residue classes ${a\ (q)}$, in the regime ${N \geq q^2}$ (say). For ${N < q^2}$, do not try to capture any cancellation and just use the triangle inequality instead.)

• (ii) Obtain a zero-free region

$\displaystyle \{ \sigma+it: \sigma > 1 - c \min( \frac{1}{(\log |t|)^{2/3} (\log\log |t|)^{1/3}}, \frac{1}{\log q} );$

$\displaystyle |t| \geq 100 \}$

for ${L(s,\chi)}$, for some (effective) absolute constant ${c>0}$.

• (iii) Obtain the prime number theorem in arithmetic progressions with error term

$\displaystyle \sum_{n \leq x: n = a\ (q)} \Lambda(n) = \frac{x}{\phi(q)} + O\left( x \exp\left( - c_A \frac{\log^{3/5} x}{(\log\log x)^{1/5}} \right) \right)$

whenever ${x > 100}$, ${q \leq \log^A x}$, ${a\ (q)}$ is primitive, and ${c_A>0}$ depends (ineffectively) on ${A}$.

We continue the discussion of sieve theory from Notes 4, but now specialise to the case of the linear sieve in which the sieve dimension ${\kappa}$ is equal to ${1}$, which is one of the best understood sieving situations, and one of the rare cases in which the precise limits of the sieve method are known. A bit more specifically, let ${z, D \geq 1}$ be quantities with ${z = D^{1/s}}$ for some fixed ${s>1}$, and let ${g}$ be a multiplicative function with

$\displaystyle g(p) = \frac{1}{p} + O(\frac{1}{p^2}) \ \ \ \ \ (1)$

and

$\displaystyle 0 \leq g(p) \leq 1-c \ \ \ \ \ (2)$

for all primes ${p}$ and some fixed ${c>0}$ (we allow all constants below to depend on ${c}$). Let ${P(z) := \prod_{p, and for each prime ${p < z}$, let ${E_p}$ be a set of integers, with ${E_d := \bigcap_{p|d} E_p}$ for ${d|P(z)}$. We consider finitely supported sequences ${(a_n)_{n \in {\bf Z}}}$ of non-negative reals for which we have bounds of the form

$\displaystyle \sum_{n \in E_d} a_n = g(d) X + r_d. \ \ \ \ \ (3)$

for all square-free ${d \leq D}$ and some ${X>0}$, and some remainder terms ${r_d}$. One is then interested in upper and lower bounds on the quantity

$\displaystyle \sum_{n\not \in\bigcup_{p

The fundamental lemma of sieve theory (Corollary 19 of Notes 4) gives us the bound

$\displaystyle \sum_{n\not \in\bigcup_{p

where ${V(z)}$ is the quantity

$\displaystyle V(z) := \prod_{p

This bound is strong when ${s}$ is large, but is not as useful for smaller values of ${s}$. We now give a sharp bound in this regime. We introduce the functions ${F, f: (0,+\infty) \rightarrow {\bf R}^+}$ by

$\displaystyle F(s) := 2e^\gamma ( \frac{1_{s>1}}{s} \ \ \ \ \ (6)$

$\displaystyle + \sum_{j \geq 3, \hbox{ odd}} \frac{1}{j!} \int_{[1,+\infty)^{j-1}} 1_{t_1+\dots+t_{j-1}\leq s-1} \frac{dt_1 \dots dt_{j-1}}{t_1 \dots t_j} )$

and

$\displaystyle f(s) := 2e^\gamma \sum_{j \geq 2, \hbox{ even}} \frac{1}{j!} \int_{[1,+\infty)^{j-1}} 1_{t_1+\dots+t_{j-1}\leq s-1} \frac{dt_1 \dots dt_{j-1}}{t_1 \dots t_j} \ \ \ \ \ (7)$

where we adopt the convention ${t_j := s - t_1 - \dots - t_{j-1}}$. Note that for each ${s}$ one has only finitely many non-zero summands in (6), (7). These functions are closely related to the Buchstab function ${\omega}$ from Exercise 28 of Supplement 4; indeed from comparing the definitions one has

$\displaystyle F(s) + f(s) = 2 e^\gamma \omega(s)$

for all ${s>0}$.

Exercise 1 (Alternate definition of ${F, f}$) Show that ${F(s)}$ is continuously differentiable except at ${s=1}$, and ${f(s)}$ is continuously differentiable except at ${s=2}$ where it is continuous, obeying the delay-differential equations

$\displaystyle \frac{d}{ds}( s F(s) ) = f(s-1) \ \ \ \ \ (8)$

for ${s > 1}$ and

$\displaystyle \frac{d}{ds}( s f(s) ) = F(s-1) \ \ \ \ \ (9)$

for ${s>2}$, with the initial conditions

$\displaystyle F(s) = \frac{2e^\gamma}{s} 1_{s>1}$

for ${s \leq 3}$ and

$\displaystyle f(s) = 0$

for ${s \leq 2}$. Show that these properties of ${F, f}$ determine ${F, f}$ completely.

For future reference, we record the following explicit values of ${F, f}$:

$\displaystyle F(s) = \frac{2e^\gamma}{s} \ \ \ \ \ (10)$

for ${1 < s \leq 3}$, and

$\displaystyle f(s) = \frac{2e^\gamma}{s} \log(s-1) \ \ \ \ \ (11)$

for ${2 \leq s \leq 4}$.

We will show

Theorem 2 (Linear sieve) Let the notation and hypotheses be as above, with ${s > 1}$. Then, for any ${\varepsilon > 0}$, one has the upper bound

$\displaystyle \sum_{n\not \in\bigcup_{p

and the lower bound

$\displaystyle \sum_{n\not \in\bigcup_{p

if ${D}$ is sufficiently large depending on ${\varepsilon, s, c}$. Furthermore, this claim is sharp in the sense that the quantity ${F(s)}$ cannot be replaced by any smaller quantity, and similarly ${f(s)}$ cannot be replaced by any larger quantity.

Comparing the linear sieve with the fundamental lemma (and also testing using the sequence ${a_n = 1_{1 \leq n \leq N}}$ for some extremely large ${N}$), we conclude that we necessarily have the asymptotics

$\displaystyle 1 - O(e^{-s}) \leq f(s) \leq 1 \leq F(s) \leq 1 + O( e^{-s} )$

for all ${s \geq 1}$; this can also be proven directly from the definitions of ${F, f}$, or from Exercise 1, but is somewhat challenging to do so; see e.g. Chapter 11 of Friedlander-Iwaniec for details.

Exercise 3 Establish the integral identities

$\displaystyle F(s) = 1 + \frac{1}{s} \int_s^\infty (1 - f(t-1))\ dt$

and

$\displaystyle f(s) = 1 + \frac{1}{s} \int_s^\infty (1 - F(t-1))\ dt$

for ${s \geq 2}$. Argue heuristically that these identities are consistent with the bounds in Theorem 2 and the Buchstab identity (Equation (16) from Notes 4).

Exercise 4 Use the Selberg sieve (Theorem 30 from Notes 4) to obtain a slightly weaker version of (12) in the range ${1 < s < 3}$ in which the error term ${|r_d|}$ is worsened to ${\tau_3(d) |r_d|}$, but the main term is unchanged.

We will prove Theorem 2 below the fold. The optimality of ${F, f}$ is closely related to the parity problem obstruction discussed in Section 5 of Notes 4; a naive application of the parity arguments there only give the weak bounds ${F(s) \geq \frac{2 e^\gamma}{s}}$ and ${f(s)=0}$ for ${s \leq 2}$, but this can be sharpened by a more careful counting of various sums involving the Liouville function ${\lambda}$.

As an application of the linear sieve (specialised to the ranges in (10), (11)), we will establish a famous theorem of Chen, giving (in some sense) the closest approach to the twin prime conjecture that one can hope to achieve by sieve-theoretic methods:

Theorem 5 (Chen’s theorem) There are infinitely many primes ${p}$ such that ${p+2}$ is the product of at most two primes.

The same argument gives the version of Chen’s theorem for the even Goldbach conjecture, namely that for all sufficiently large even ${N}$, there exists a prime ${p}$ between ${2}$ and ${N}$ such that ${N-p}$ is the product of at most two primes.

The discussion in these notes loosely follows that of Friedlander-Iwaniec (who study sieving problems in more general dimension than ${\kappa=1}$).