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Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and I have just uploaded to the arXiv our paper “Long gaps between primes“. This is a followup work to our two previous papers (discussed in this previous post), in which we had simultaneously shown that the maximal gap

between primes up to exhibited a lower bound of the shape

for some function that went to infinity as ; this improved upon previous work of Rankin and other authors, who established the same bound but with replaced by a constant. (Again, see the previous post for a more detailed discussion.)

In our previous papers, we did not specify a particular growth rate for . In my paper with Kevin, Ben, and Sergei, there was a good reason for this: our argument relied (amongst other things) on the inverse conjecture on the Gowers norms, as well as the Siegel-Walfisz theorem, and the known proofs of both results both have ineffective constants, rendering our growth function similarly ineffective. Maynard’s approach ostensibly also relies on the Siegel-Walfisz theorem, but (as shown in another recent paper of his) can be made quite effective, even when tracking -tuples of fairly large size (about for some small ). If one carefully makes all the bounds in Maynard’s argument quantitative, one eventually ends up with a growth rate of shape

on the gaps between primes for large ; this is an unpublished calculation of James’.

In this paper we make a further refinement of this calculation to obtain a growth rate

leading to a bound of the form

for large and some small constant . Furthermore, this appears to be the limit of current technology (in particular, falling short of Cramer’s conjecture that is comparable to ); in the spirit of Erdös’ original prize on this problem, I would like to offer 10,000 USD for anyone who can show (in a refereed publication, of course) that the constant here can be replaced by an arbitrarily large constant .

The reason for the growth rate (3) is as follows. After following the sieving process discussed in the previous post, the problem comes down to something like the following: can one sieve out all (or almost all) of the primes in by removing one residue class modulo for all primes in (say) ? Very roughly speaking, if one can solve this problem with , then one can obtain a growth rate on of the shape . (This is an oversimplification, as one actually has to sieve out a random subset of the primes, rather than all the primes in , but never mind this detail for now.)

Using the quantitative “dense clusters of primes” machinery of Maynard, one can find lots of -tuples in which contain at least primes, for as large as or so (so that is about ). By considering -tuples in arithmetic progression, this means that one can find lots of residue classes modulo a given prime in that capture about primes. In principle, this means that union of all these residue classes can cover about primes, allowing one to take as large as , which corresponds to (3). However, there is a catch: the residue classes for different primes may collide with each other, reducing the efficiency of the covering. In our previous papers on the subject, we selected the residue classes randomly, which meant that we had to insert an additional logarithmic safety margin in expected number of times each prime would be shifted out by one of the residue classes, in order to guarantee that we would (with high probability) sift out most of the primes. This additional safety margin is ultimately responsible for the loss in (2).

The main innovation of this paper, beyond detailing James’ unpublished calculations, is to use ideas from the literature on efficient hypergraph covering, to avoid the need for a logarithmic safety margin. The hypergraph covering problem, roughly speaking, is to try to cover a set of vertices using as few “edges” from a given hypergraph as possible. If each edge has vertices, then one certainly needs at least edges to cover all the vertices, and the question is to see if one can come close to attaining this bound given some reasonable uniform distribution hypotheses on the hypergraph . As before, random methods tend to require something like edges before one expects to cover, say of the vertices.

However, it turns out (under reasonable hypotheses on ) to eliminate this logarithmic loss, by using what is now known as the “semi-random method” or the “Rödl nibble”. The idea is to randomly select a small number of edges (a first “nibble”) – small enough that the edges are unlikely to overlap much with each other, thus obtaining maximal efficiency. Then, one pauses to remove all the edges from that intersect edges from this first nibble, so that all remaining edges will not overlap with the existing edges. One then randomly selects another small number of edges (a second “nibble”), and repeats this process until enough nibbles are taken to cover most of the vertices. Remarkably, it turns out that under some reasonable assumptions on the hypergraph , one can maintain control on the uniform distribution of the edges throughout the nibbling process, and obtain an efficient hypergraph covering. This strategy was carried out in detail in an influential paper of Pippenger and Spencer.

In our setup, the vertices are the primes in , and the edges are the intersection of the primes with various residue classes. (Technically, we have to work with a family of hypergraphs indexed by a prime , rather than a single hypergraph, but let me ignore this minor technical detail.) The semi-random method would *in principle* eliminate the logarithmic loss and recover the bound (3). However, there is a catch: the analysis of Pippenger and Spencer relies heavily on the assumption that the hypergraph is uniform, that is to say all edges have the same size. In our context, this requirement would mean that each residue class captures exactly the same number of primes, which is not the case; we only control the number of primes in an average sense, but we were unable to obtain any concentration of measure to come close to verifying this hypothesis. And indeed, the semi-random method, when applied naively, does not work well with edges of variable size – the problem is that edges of large size are much more likely to be eliminated after each nibble than edges of small size, since they have many more vertices that could overlap with the previous nibbles. Since the large edges are clearly the more useful ones for the covering problem than small ones, this bias towards eliminating large edges significantly reduces the efficiency of the semi-random method (and also greatly complicates the *analysis* of that method).

Our solution to this is to iteratively *reweight* the probability distribution on edges after each nibble to compensate for this bias effect, giving larger edges a greater weight than smaller edges. It turns out that there is a natural way to do this reweighting that allows one to repeat the Pippenger-Spencer analysis in the presence of edges of variable size, and this ultimately allows us to recover the full growth rate (3).

To go beyond (3), one either has to find a lot of residue classes that can capture significantly more than primes of size (which is the limit of the multidimensional Selberg sieve of Maynard and myself), or else one has to find a very different method to produce large gaps between primes than the Erdös-Rankin method, which is the method used in all previous work on the subject.

It turns out that the arguments in this paper can be combined with the Maier matrix method to also produce chains of consecutive large prime gaps whose size is of the order of (4); three of us (Kevin, James, and myself) will detail this in a future paper. (A similar combination was also recently observed in connection with our earlier result (1) by Pintz, but there are some additional technical wrinkles required to recover the full gain of (3) for the chains of large gaps problem.)

In Notes 2, the Riemann zeta function (and more generally, the Dirichlet -functions ) were extended meromorphically into the region in and to the right of the critical strip. This is a sufficient amount of meromorphic continuation for many applications in analytic number theory, such as establishing the prime number theorem and its variants. The zeroes of the zeta function in the critical strip are known as the *non-trivial zeroes* of , and thanks to the truncated explicit formulae developed in Notes 2, they control the asymptotic distribution of the primes (up to small errors).

The function obeys the trivial functional equation

for all in its domain of definition. Indeed, as is real-valued when is real, the function vanishes on the real line and is also meromorphic, and hence vanishes everywhere. Similarly one has the functional equation

From these equations we see that the zeroes of the zeta function are symmetric across the real axis, and the zeroes of are the reflection of the zeroes of across this axis.

It is a remarkable fact that these functions obey an additional, and more non-trivial, functional equation, this time establishing a symmetry across the *critical line* rather than the real axis. One consequence of this symmetry is that the zeta function and -functions may be extended meromorphically to the entire complex plane. For the zeta function, the functional equation was discovered by Riemann, and reads as follows:

Theorem 1 (Functional equation for the Riemann zeta function)The Riemann zeta function extends meromorphically to the entire complex plane, with a simple pole at and no other poles. Furthermore, one has the functional equationfor all complex other than , where is the function

Here , are the complex-analytic extensions of the classical trigionometric functions , and is the Gamma function, whose definition and properties we review below the fold.

The functional equation can be placed in a more symmetric form as follows:

Corollary 2 (Functional equation for the Riemann xi function)The Riemann xi functionis analytic on the entire complex plane (after removing all removable singularities), and obeys the functional equations

In particular, the zeroes of consist precisely of the non-trivial zeroes of , and are symmetric about both the real axis and the critical line. Also, is real-valued on the critical line and on the real axis.

Corollary 2 is an easy consequence of Theorem 1 together with the duplication theorem for the Gamma function, and the fact that has no zeroes to the right of the critical strip, and is left as an exercise to the reader (Exercise 19). The functional equation in Theorem 1 has many proofs, but most of them are related in on way or another to the Poisson summation formula

(Theorem 34 from Supplement 2, at least in the case when is twice continuously differentiable and compactly supported), which can be viewed as a Fourier-analytic link between the coarse-scale distribution of the integers and the fine-scale distribution of the integers. Indeed, there is a quick heuristic proof of the functional equation that comes from formally applying the Poisson summation formula to the function , and noting that the functions and are formally Fourier transforms of each other, up to some Gamma function factors, as well as some trigonometric factors arising from the distinction between the real line and the half-line. Such a heuristic proof can indeed be made rigorous, and we do so below the fold, while also providing Riemann’s two classical proofs of the functional equation.

From the functional equation (and the poles of the Gamma function), one can see that has *trivial zeroes* at the negative even integers , in addition to the non-trivial zeroes in the critical strip. More generally, the following table summarises the zeroes and poles of the various special functions appearing in the functional equation, after they have been meromorphically extended to the entire complex plane, and with zeroes classified as “non-trivial” or “trivial” depending on whether they lie in the critical strip or not. (Exponential functions such as or have no zeroes or poles, and will be ignored in this table; the zeroes and poles of rational functions such as are self-evident and will also not be displayed here.)

Function | Non-trivial zeroes | Trivial zeroes | Poles |

Yes | |||

Yes | |||

No | Even integers | No | |

No | Odd integers | No | |

No | Integers | No | |

No | No | ||

No | No | ||

No | No | ||

No | No | ||

Yes | No | No |

Among other things, this table indicates that the Gamma and trigonometric factors in the functional equation are tied to the trivial zeroes and poles of zeta, but have no direct bearing on the distribution of the non-trivial zeroes, which is the most important feature of the zeta function for the purposes of analytic number theory, beyond the fact that they are symmetric about the real axis and critical line. In particular, the Riemann hypothesis is not going to be resolved just from further analysis of the Gamma function!

The zeta function computes the “global” sum , with ranging all the way from to infinity. However, by some Fourier-analytic (or complex-analytic) manipulation, it is possible to use the zeta function to also control more “localised” sums, such as for some and some smooth compactly supported function . It turns out that the functional equation (3) for the zeta function localises to this context, giving an *approximate functional equation* which roughly speaking takes the form

whenever and ; see Theorem 38 below for a precise formulation of this equation. Unsurprisingly, this form of the functional equation is also very closely related to the Poisson summation formula (8), indeed it is essentially a special case of that formula (or more precisely, of the van der Corput -process). This useful identity relates long smoothed sums of to short smoothed sums of (or vice versa), and can thus be used to shorten exponential sums involving terms such as , which is useful when obtaining some of the more advanced estimates on the Riemann zeta function.

We will give two other basic uses of the functional equation. The first is to get a good count (as opposed to merely an upper bound) on the density of zeroes in the critical strip, establishing the Riemann-von Mangoldt formula that the number of zeroes of imaginary part between and is for large . The other is to obtain untruncated versions of the explicit formula from Notes 2, giving a remarkable exact formula for sums involving the von Mangoldt function in terms of zeroes of the Riemann zeta function. These results are not strictly necessary for most of the material in the rest of the course, but certainly help to clarify the nature of the Riemann zeta function and its relation to the primes.

In view of the material in previous notes, it should not be surprising that there are analogues of all of the above theory for Dirichlet -functions . We will restrict attention to primitive characters , since the -function for imprimitive characters merely differs from the -function of the associated primitive factor by a finite Euler product; indeed, if for some principal whose modulus is coprime to that of , then

(cf. equation (45) of Notes 2).

The main new feature is that the Poisson summation formula needs to be “twisted” by a Dirichlet character , and this boils down to the problem of understanding the finite (additive) Fourier transform of a Dirichlet character. This is achieved by the classical theory of Gauss sums, which we review below the fold. There is one new wrinkle; the value of plays a role in the functional equation. More precisely, we have

Theorem 3 (Functional equation for -functions)Let be a primitive character of modulus with . Then extends to an entire function on the complex plane, withor equivalently

for all , where is equal to in the even case and in the odd case , and

where is the Gauss sum

and , with the convention that the -periodic function is also (by abuse of notation) applied to in the cyclic group .

From this functional equation and (2) we see that, as with the Riemann zeta function, the non-trivial zeroes of (defined as the zeroes within the critical strip are symmetric around the critical line (and, if is real, are also symmetric around the real axis). In addition, acquires trivial zeroes at the negative even integers and at zero if , and at the negative odd integers if . For imprimitive , we see from (9) that also acquires some additional trivial zeroes on the left edge of the critical strip.

There is also a symmetric version of this equation, analogous to Corollary 2:

Corollary 4Let be as above, and setthen is entire with .

For further detail on the functional equation and its implications, I recommend the classic text of Titchmarsh or the text of Davenport.

In Notes 1, we approached multiplicative number theory (the study of multiplicative functions and their relatives) via elementary methods, in which attention was primarily focused on obtaining asymptotic control on summatory functions and logarithmic sums . Now we turn to the complex approach to multiplicative number theory, in which the focus is instead on obtaining various types of control on the Dirichlet series , defined (at least for of sufficiently large real part) by the formula

These series also made an appearance in the elementary approach to the subject, but only for real that were larger than . But now we will exploit the freedom to extend the variable to the complex domain; this gives enough freedom (in principle, at least) to recover control of elementary sums such as or from control on the Dirichlet series. Crucially, for many key functions of number-theoretic interest, the Dirichlet series can be analytically (or at least meromorphically) continued to the left of the line . The zeroes and poles of the resulting meromorphic continuations of (and of related functions) then turn out to control the asymptotic behaviour of the elementary sums of ; the more one knows about the former, the more one knows about the latter. In particular, knowledge of where the zeroes of the Riemann zeta function are located can give very precise information about the distribution of the primes, by means of a fundamental relationship known as the explicit formula. There are many ways of phrasing this explicit formula (both in exact and in approximate forms), but they are all trying to formalise an approximation to the von Mangoldt function (and hence to the primes) of the form

where the sum is over zeroes (counting multiplicity) of the Riemann zeta function (with the sum often restricted so that has large real part and bounded imaginary part), and the approximation is in a suitable weak sense, so that

for suitable “test functions” (which in practice are restricted to be fairly smooth and slowly varying, with the precise amount of restriction dependent on the amount of truncation in the sum over zeroes one wishes to take). Among other things, such approximations can be used to rigorously establish the prime number theorem

as , with the size of the error term closely tied to the location of the zeroes of the Riemann zeta function.

The explicit formula (1) (or any of its more rigorous forms) is closely tied to the counterpart approximation

for the Dirichlet series of the von Mangoldt function; note that (4) is formally the special case of (2) when . Such approximations come from the general theory of local factorisations of meromorphic functions, as discussed in Supplement 2; the passage from (4) to (2) is accomplished by such tools as the residue theorem and the Fourier inversion formula, which were also covered in Supplement 2. The relative ease of uncovering the Fourier-like duality between primes and zeroes (sometimes referred to poetically as the “music of the primes”) is one of the major advantages of the complex-analytic approach to multiplicative number theory; this important duality tends to be rather obscured in the other approaches to the subject, although it can still in principle be discernible with sufficient effort.

More generally, one has an explicit formula

for any Dirichlet character , where now ranges over the zeroes of the associated Dirichlet -function ; we view this formula as a “twist” of (1) by the Dirichlet character . The explicit formula (5), proven similarly (in any of its rigorous forms) to (1), is important in establishing the prime number theorem in arithmetic progressions, which asserts that

as , whenever is a fixed primitive residue class. Again, the size of the error term here is closely tied to the location of the zeroes of the Dirichlet -function, with particular importance given to whether there is a zero very close to (such a zero is known as an *exceptional zero* or Siegel zero).

While any information on the behaviour of zeta functions or -functions is in principle welcome for the purposes of analytic number theory, some regions of the complex plane are more important than others in this regard, due to the differing weights assigned to each zero in the explicit formula. Roughly speaking, in descending order of importance, the most crucial regions on which knowledge of these functions is useful are

- The region on or near the point .
- The region on or near the right edge of the
*critical strip*. - The right half of the critical strip.
- The region on or near the
*critical line*that bisects the critical strip. - Everywhere else.

For instance:

- We will shortly show that the Riemann zeta function has a simple pole at with residue , which is already sufficient to recover much of the classical theorems of Mertens discussed in the previous set of notes, as well as results on mean values of multiplicative functions such as the divisor function . For Dirichlet -functions, the behaviour is instead controlled by the quantity discussed in Notes 1, which is in turn closely tied to the existence and location of a Siegel zero.
- The zeta function is also known to have no zeroes on the right edge of the critical strip, which is sufficient to prove (and is in fact equivalent to) the prime number theorem. Any enlargement of the zero-free region for into the critical strip leads to improved error terms in that theorem, with larger zero-free regions leading to stronger error estimates. Similarly for -functions and the prime number theorem in arithmetic progressions.
- The (as yet unproven) Riemann hypothesis prohibits from having any zeroes within the right half of the critical strip, and gives very good control on the number of primes in intervals, even when the intervals are relatively short compared to the size of the entries. Even without assuming the Riemann hypothesis,
*zero density estimates*in this region are available that give some partial control of this form. Similarly for -functions, primes in short arithmetic progressions, and the generalised Riemann hypothesis. - Assuming the Riemann hypothesis, further distributional information about the zeroes on the critical line (such as Montgomery’s pair correlation conjecture, or the more general
*GUE hypothesis*) can give finer information about the error terms in the prime number theorem in short intervals, as well as other arithmetic information. Again, one has analogues for -functions and primes in short arithmetic progressions. - The functional equation of the zeta function describes the behaviour of to the left of the critical line, in terms of the behaviour to the right of the critical line. This is useful for building a “global” picture of the structure of the zeta function, and for improving a number of estimates about that function, but (in the absence of unproven conjectures such as the Riemann hypothesis or the pair correlation conjecture) it turns out that many of the basic analytic number theory results using the zeta function can be established without relying on this equation. Similarly for -functions.

Remark 1If one takes an “adelic” viewpoint, one can unite the Riemann zeta function and all of the -functions for various Dirichlet characters into a single object, viewing as a general multiplicative character on the adeles; thus the imaginary coordinate and the Dirichlet character are really the Archimedean and non-Archimedean components respectively of a single adelic frequency parameter. This viewpoint was famously developed in Tate’s thesis, which among other things helps to clarify the nature of the functional equation, as discussed in this previous post. We will not pursue the adelic viewpoint further in these notes, but it does supply a “high-level” explanation for why so much of the theory of the Riemann zeta function extends to the Dirichlet -functions. (The non-Archimedean character and the Archimedean character behave similarly from an algebraic point of view, but not so much from an analytic point of view; as such, the adelic viewpoint is well suited for algebraic tasks (such as establishing the functional equation), but not for analytic tasks (such as establishing a zero-free region).)

Roughly speaking, the elementary multiplicative number theory from Notes 1 corresponds to the information one can extract from the complex-analytic method in region 1 of the above hierarchy, while the more advanced elementary number theory used to prove the prime number theorem (and which we will not cover in full detail in these notes) corresponds to what one can extract from regions 1 and 2.

As a consequence of this hierarchy of importance, information about the function away from the critical strip, such as Euler’s identity

or equivalently

or the infamous identity

which is often presented (slightly misleadingly, if one’s conventions for divergent summation are not made explicit) as

are of relatively little direct importance in analytic prime number theory, although they are still of interest for some other, non-number-theoretic, applications. (The quantity does play a minor role as a normalising factor in some asymptotics, see e.g. Exercise 28 from Notes 1, but its precise value is usually not of major importance.) In contrast, the value of an -function at turns out to be extremely important in analytic number theory, with many results in this subject relying ultimately on a non-trivial lower-bound on this quantity coming from Siegel’s theorem, discussed below the fold.

For a more in-depth treatment of the topics in this set of notes, see Davenport’s “Multiplicative number theory“.

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

Exercise 1Show that the field of algebraic numbers is indeed a field, and that the ring of algebraic integers is indeed a ring, and is in fact an integral domain. Also, show that , that is to say the ordinary integers are precisely the algebraic integers that are also rational. Because of this, we will sometimes refer to elements of asrational integers.

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

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

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

Exercise 3Let be a square-free natural number with or . Show that the ring of algebraic integers in is given byIf instead is square-free with , show that the ring is instead given by

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

Remark 4In the case , it may naively appear more natural to work with the ring , which is an index two subring of . However, because this ring only captures some of the algebraic integers in rather than all of them, the algebraic properties of these rings are somewhat worse than those of (in particular, they generally fail to be Dedekind domains) and so are not convenient to work with in algebraic number theory.

We refer to fields of the form for natural square-free numbers as *quadratic fields of negative discriminant*, and similarly refer to as a ring of quadratic integers of negative discriminant. Quadratic fields and quadratic integers of positive discriminant are just as important to analytic number theory as their negative discriminant counterparts, but we will restrict attention to the latter here for simplicity of discussion.

Thus, for instance, when , the ring of integers in is the ring of Gaussian integers

and when , the ring of integers in is the ring of Eisenstein integers

where is a cube root of unity.

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

For instance, every rational integer has an absolute value , with the multiplicativity property for , and the positivity property for all . Among other things, the absolute value detects units: if and only if is a unit in (that is to say, it is multiplicatively invertible in ). Similarly, in any ring of quadratic integers with negative discriminant, we can assign a norm to any quadratic integer by the formula

where is the complex conjugate of . (When working with other number fields than quadratic fields of negative discriminant, one instead defines to be the product of all the Galois conjugates of .) Thus for instance, when one has

Analogously to the rational integers, we have the multiplicativity property for and the positivity property for , and the units in are precisely the elements of norm one.

Exercise 5Establish the three claims of the previous paragraph. Conclude that the units (invertible elements) of consist of the four elements if , the six elements if , and the two elements if .

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

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

The norm forms (1), (2) can be viewed as examples of positive definite quadratic forms over the integers, by which we mean a polynomial of the form

for some integer coefficients . One can declare two quadratic forms to be *equivalent* if one can transform one to the other by an invertible linear transformation , so that . For example, the quadratic forms and are equivalent, as can be seen by using the invertible linear transformation . Such equivalences correspond to the different choices of basis available when expressing a ring such as (or an ideal thereof) additively as a copy of .

There is an important and classical invariant of a quadratic form , namely the discriminant , which will of course be familiar to most readers via the quadratic formula, which among other things tells us that a quadratic form will be positive definite precisely when its discriminant is negative. It is not difficult (particularly if one exploits the multiplicativity of the determinant of matrices) to show that two equivalent quadratic forms have the same discriminant. Thus for instance any quadratic form equivalent to (1) has discriminant , while any quadratic form equivalent to (2) has discriminant . Thus we see that each ring of quadratic integers is associated with a certain negative discriminant , defined to equal when and when .

Exercise 6 (Geometric interpretation of discriminant)Let be a quadratic form of negative discriminant , and extend it to a real form in the obvious fashion. Show that for any , the set is an ellipse of area .

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

Exercise 7Show that any quadratic form of discriminant is equivalent to the form , and any quadratic form of discriminant is equivalent to . (Hint:use elementary transformations to try to make as small as possible, to the point where one only has to check a finite number of cases; this argument is due to Legendre.) More generally, show that for any negative discriminant , there are only finitely many quadratic forms of that discriminant up to equivalence (a result first established by Gauss).

Unfortunately, for most choices of discriminant, the converse question fails; for instance, the quadratic forms and both have discriminant , but are not equivalent (Exercise 38). This particular failure of equivalence turns out to be intimately related to the failure of unique factorisation in the ring .

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

Proposition 8Let be the real non-principal Dirichlet character of modulus , or more explicitly is equal to when , when , and when .

- (i) For any natural number , the number of Gaussian integers with norm is equal to . Equivalently, the number of solutions to the equation with is . (Here, as in the previous post, the symbol denotes Dirichlet convolution.)
- (ii) For any natural number , the number of Gaussian integers that divide (thus for some ) is .

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

As an illustration of the relevance of such connections to analytic number theory, let us now explicitly compute .

This particular identity is also known as the Leibniz formula.

*Proof:* For a large number , consider the quantity

of all the Gaussian integers of norm less than . On the one hand, this is the same as the number of lattice points of in the disk of radius . Placing a unit square centred at each such lattice point, we obtain a region which differs from the disk by a region contained in an annulus of area . As the area of the disk is , we conclude the Gauss bound

On the other hand, by Proposition 8(i) (and removing the contribution), we see that

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

and then using the bounds , , from the previous set of notes to conclude that

Comparing the two formulae for and sending , we obtain the claim.

Exercise 10Give an alternate proof of Corollary 9 that relies on obtaining asymptotics for the Dirichlet series as , rather than using the Dirichlet hyperbola method.

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

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

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

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

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

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

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

- The
*summatory functions*of an arithmetic function , as well as the associated natural density

(if it exists).

- The
*logarithmic sums*of an arithmetic function , as well as the associated

*logarithmic density*(if it exists).

Here, we are normalising the arithmetic function being studied to be of roughly unit size up to logarithms, obeying bounds such as , , or at worst

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

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

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

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

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

Remark 1The elementary and complex-analytic approaches to multiplicative number theory are the two classical approaches to the subject. One could also consider a more “Fourier-analytic” approach, in which one studies convolution-type statistics such asas for various cutoff functions , such as smooth, compactly supported functions. See for instance this previous blog post for an instance of such an approach. Another related approach is the “pretentious” approach to multiplicative number theory currently being developed by Granville-Soundararajan and their collaborators. We will occasionally make reference to these more modern approaches in these notes, but will primarily focus on the classical approaches.

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

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

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

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

- For the twin prime conjecture, one can use the linear forms , , and the property in question is the assertion that and are both prime.
- For the even Goldbach conjecture, the claim is similar but one uses the linear forms , for some even integer .
- For Chen’s theorem, we use the same linear forms as in the previous two cases, but now is the assertion that is prime and is an almost prime (in the sense that there are at most two prime factors).
- In the recent results establishing bounded gaps between primes, we use the linear forms for some admissible tuple , and take to be the assertion that at least two of are prime.

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

- First, one chooses a carefully selected
*sieve weight*, which could for instance be a non-negative function having a divisor sum formfor some coefficients , where is a natural scale parameter. The precise choice of sieve weight is often quite a delicate matter, but will not be discussed here. (In some cases, one may work with multiple sieve weights .)

- Next, one uses tools from analytic number theory (such as the Bombieri-Vinogradov theorem) to obtain upper and lower bounds for sums such as
where is some “arithmetic” function involving the prime factorisation of (we will be a bit vague about what this means precisely, but a typical choice of might be a Dirichlet convolution of two other arithmetic functions ).

- Using some combinatorial arguments, one manipulates these upper and lower bounds, together with the non-negative nature of , to conclude the existence of an in the support of (or of at least one of the sieve weights being considered) for which holds

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

and lower bounds on

so that one can show that the expression

is strictly positive, which implies the existence of an in the support of such that at least two of are prime. As another example, to prove Chen’s theorem to find such that is prime and is almost prime, one uses a variety of sieve weights to produce a lower bound for

and an upper bound for

and

where is some parameter between and , and “rough” means that all prime factors are at least . One can observe that if , then there must be at least one for which is prime and is almost prime, since for any rough number , the quantity

is only positive when is an almost prime (if has three or more factors, then either it has at least two factors less than , or it is of the form for some ). The upper and lower bounds on are ultimately produced via asymptotics for expressions of the form (1), (2), (3) for various divisor sums and various arithmetic functions .

Unfortunately, there is an obstruction to sieve-theoretic techniques working for certain types of properties , which Zeb Brady and I recently formalised at an AIM workshop this week. To state the result, we recall the Liouville function , defined by setting whenever is the product of exactly primes (counting multiplicity). Define a *sign pattern* to be an element of the discrete cube . Given a property of natural numbers , we say that a sign pattern is *forbidden* by if there does not exist any natural numbers obeying for which

Example 1Let be the property that at least two of are prime. Then the sign patterns , , , are forbidden, because prime numbers have a Liouville function of , so that can only occur when at least two of are equal to .

Example 2Let be the property that is prime and is almost prime. Then the only forbidden sign patterns are and .

Example 3Let be the property that and are both prime. Then are all forbidden sign patterns.

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

Claim 1 (Parity obstruction)Suppose is such that that the convex hull of the forbidden sign patterns of contains the origin. Then one cannot use the above sieve-theoretic approach to establish the existence of an such that holds.

Thus for instance, the property in Example 3 is subject to the parity obstruction since is a convex combination of and , whereas the properties in Examples 1, 2 are not. One can also check that the property “at least of the numbers is prime” is subject to the parity obstruction as soon as . Thus, the largest number of elements of a -tuple that one can force to be prime by purely sieve-theoretic methods is , rounded up.

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

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

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

for all . By Fourier expansion (or Lagrange interpolation), one can then write as a polynomial

where is a polynomial in variables that is a linear combination of monomials with and (thus has no constant or linear terms, and no monomials with repeated terms). The point is that the mean zero condition allows one to eliminate the linear terms. If we now consider the weight function

then is non-negative, is supported solely on for which is a forbidden pattern, and is equal to plus a linear combination of monomials with .

The Liouville pseudorandomness principle then predicts that sums of the form

and

or more generally

should be asymptotically negligible; intuitively, the point here is that the prime factorisation of should not influence the Liouville function of , even on the short arithmetic progressions that the divisor sum is built out of, and so any monomial occurring in should exhibit strong cancellation for any of the above sums. If one accepts this principle, then all the expressions (1), (2), (3) should be essentially unchanged when is replaced by .

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

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

for all forbidden sign patterns and some . On the other hand, from Liouville pseudorandomness one expects that

is negligible (as compared against for any reasonable sieve weight . We conclude that for some in the support of , that

and hence is not a forbidden sign pattern. This does not actually imply that holds, but it does not prevent from holding purely from parity considerations. Thus, we do not expect a parity obstruction of the type in Claim 1 to hold when the convex hull of forbidden sign patterns does not contain the origin.

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

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

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

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

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

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

for any fixed . Unconditionally, the best result so far (up to logarithmic factors) that holds for all primes is by Burgess, who showed that

for any fixed . See this previous post for a proof of these bounds.

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

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

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

must be unusually large also. However, because is large, one can factorise as for a fairly sparsely supported function . The Elliott-Halberstam conjecture, which controls the distribution of in arithmetic progressions on the average can then be used to show that is small, giving the required contradiction.

The implication involving Type II estimates is proven by a variant of the argument. If is large, then a character sum is unusually large for a certain . By multiplicativity of , this shows that correlates with , and then by periodicity of , this shows that correlates with for various small . By the Cauchy-Schwarz inequality (cf. this previous blog post), this implies that correlates with for some distinct . But this can be ruled out by using Type II estimates.

I’ll record here a well-known observation concerning potential counterexamples to any improvement to the Burgess bound, that is to say an infinite sequence of primes with . Suppose we let be the asymptotic mean value of the quadratic character at and the mean value of ; these quantities are defined precisely in my paper, but roughly speaking one can think of

and

Thanks to the basic Dirichlet convolution identity , one can establish the *Wirsing integral equation*

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

while from the Burgess exponential sum estimates we have

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

and

and then for , evolves by the integral equation

For instance

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

The prime number theorem can be expressed as the assertion

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

where the second von Mangoldt function is defined by the formula

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

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

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

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

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

Theorem 1 (Construction of a Banach algebra norm)For any , let denote the quantityThen is a seminorm on with the bound

for all . Furthermore, we have the Banach algebra bound

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

Theorem 2 (Non-trivial Banach algebras with many local units have non-trivial spectrum)Let be a seminorm on obeying (7), (8). Suppose that is not identically zero. Then there exists such thatfor all . In particular, by (7), one has

whenever is a non-negative function.

The second is a consequence of the Selberg symmetry formula and the fact that is real (as well as Mertens’ theorem, in the case), and is closely related to the non-vanishing of the Riemann zeta function on the line :

Theorem 3 (Breaking the parity barrier)Let . Then there exists such that is non-negative, and

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

as for any Schwartz function (the decay rate in may depend on ). Specialising to functions of the form for some smooth compactly supported on , we conclude that

as ; by the smooth Urysohn lemma this implies that

as for any fixed , and the prime number theorem then follows by a telescoping series argument.

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

for any fixed Dirichlet character ; the one difference is that the use of Mertens’ theorem is replaced by the basic fact that the quantity is non-vanishing.

Analytic number theory is often concerned with the asymptotic behaviour of various arithmetic functions: functions or from the natural numbers to the real numbers or complex numbers . In this post, we will focus on the purely algebraic properties of these functions, and for reasons that will become clear later, it will be convenient to generalise the notion of an arithmetic function to functions taking values in some abstract commutative ring . In this setting, we can add or multiply two arithmetic functions to obtain further arithmetic functions , and we can also form the Dirichlet convolution by the usual formula

Regardless of what commutative ring is in used here, we observe that Dirichlet convolution is commutative, associative, and bilinear over .

An important class of arithmetic functions in analytic number theory are the multiplicative functions, that is to say the arithmetic functions such that and

for all coprime . A subclass of these functions are the completely multiplicative functions, in which the restriction that be coprime is dropped. Basic examples of completely multiplicative functions (in the classical setting ) include

- the Kronecker delta , defined by setting for and otherwise;
- the constant function and the linear function (which by abuse of notation we denote by );
- more generally monomials for any fixed complex number (in particular, the “Archimedean characters” for any fixed ), which by abuse of notation we denote by ;
- Dirichlet characters ;
- the Liouville function ;
- the indicator function of the -smooth numbers (numbers whose prime factors are all at most ), for some given ; and
- the indicator function of the -rough numbers (numbers whose prime factors are all greater than ), for some given .

Examples of multiplicative functions that are not completely multiplicative include

- the Möbius function ;
- the divisor function (also referred to as );
- more generally, the higher order divisor functions for ;
- the Euler totient function ;
- the number of roots of a given polynomial defined over ;
- more generally, the point counting function of a given algebraic variety defined over (closely tied to the Hasse-Weil zeta function of );
- the function that counts the number of representations of as the sum of two squares;
- more generally, the function that maps a natural number to the number of ideals in a given number field of absolute norm (closely tied to the Dedekind zeta function of ).

These multiplicative functions interact well with the multiplication and convolution operations: if are multiplicative, then so are and , and if is completely multiplicative, then we also have

Finally, the product of completely multiplicative functions is again completely multiplicative. On the other hand, the sum of two multiplicative functions will never be multiplicative (just look at what happens at ), and the convolution of two completely multiplicative functions will usually just be multiplicative rather than completley multiplicative.

The specific multiplicative functions listed above are also related to each other by various important identities, for instance

where is an arbitrary arithmetic function.

On the other hand, analytic number theory also is very interested in certain arithmetic functions that are *not* exactly multiplicative (and certainly not completely multiplicative). One particularly important such function is the von Mangoldt function . This function is certainly not multiplicative, but is clearly closely related to such functions via such identities as and , where is the natural logarithm function. The purpose of this post is to point out that functions such as the von Mangoldt function lie in a class closely related to multiplicative functions, which I will call the *derived multiplicative functions*. More precisely:

Definition 1Aderived multiplicative functionis an arithmetic function that can be expressed as the formal derivativeat the origin of a family of multiplicative functions parameterised by a formal parameter . Equivalently, is a derived multiplicative function if it is the coefficient of a multiplicative function in the extension of by a nilpotent infinitesimal ; in other words, there exists an arithmetic function such that the arithmetic function is multiplicative, or equivalently that is multiplicative and one has the Leibniz rule

More generally, for any , a

-derived multiplicative functionis an arithmetic function that can be expressed as the formal derivativeat the origin of a family of multiplicative functions parameterised by formal parameters . Equivalently, is the coefficient of a multiplicative function in the extension of by nilpotent infinitesimals .

We define the notion of a -derived completely multiplicative function similarly by replacing “multiplicative” with “completely multiplicative” in the above discussion.

There are Leibniz rules similar to (2) but they are harder to state; for instance, a doubly derived multiplicative function comes with singly derived multiplicative functions and a multiplicative function such that

for all coprime .

One can then check that the von Mangoldt function is a derived multiplicative function, because is multiplicative in the ring with one infinitesimal . Similarly, the logarithm function is derived completely multiplicative because is completely multiplicative in . More generally, any additive function is derived multiplicative because it is the top order coefficient of .

Remark 1One can also phrase these concepts in terms of the formal Dirichlet series associated to an arithmetic function . A function is multiplicative if admits a (formal) Euler product; is derived multiplicative if is the (formal) first derivative of an Euler product with respect to some parameter (not necessarily , although this is certainly an option); and so forth.

Using the definition of a -derived multiplicative function as the top order coefficient of a multiplicative function of a ring with infinitesimals, it is easy to see that the product or convolution of a -derived multiplicative function and a -derived multiplicative function is necessarily a -derived multiplicative function (again taking values in ). Thus, for instance, the higher-order von Mangoldt functions are -derived multiplicative functions, because is a -derived completely multiplicative function. More explicitly, is the top order coeffiicent of the completely multiplicative function , and is the top order coefficient of the multiplicative function , with both functions taking values in the ring of complex numbers with infinitesimals attached.

It then turns out that most (if not all) of the basic identities used by analytic number theorists concerning derived multiplicative functions, can in fact be viewed as coefficients of identities involving purely multiplicative functions, with the latter identities being provable primarily from multiplicative identities, such as (1). This phenomenon is analogous to the one in linear algebra discussed in this previous blog post, in which many of the trace identities used there are derivatives of determinant identities. For instance, the Leibniz rule

for any arithmetic functions can be viewed as the top order term in

in the ring with one infinitesimal , and then we see that the Leibniz rule is a special case (or a derivative) of (1), since is completely multiplicative. Similarly, the formulae

are top order terms of

and the variant formula is the top order term of

which can then be deduced from the previous identities by noting that the completely multiplicative function inverts multiplicatively, and also noting that annihilates . The Selberg symmetry formula

which plays a key role in the Erdös-Selberg elementary proof of the prime number theorem (as discussed in this previous blog post), is the top order term of the identity

involving the multiplicative functions , , , with two infinitesimals , and this identity can be proven while staying purely within the realm of multiplicative functions, by using the identities

and (1). Similarly for higher identities such as

which arise from expanding out using (1) and the above identities; we leave this as an exercise to the interested reader.

An analogous phenomenon arises for identities that are not purely multiplicative in nature due to the presence of truncations, such as the Vaughan identity

for any , where is the restriction of a multiplicative function to the natural numbers greater than , and similarly for , , . In this particular case, (4) is the top order coefficient of the identity

which can be easily derived from the identities and . Similarly for the Heath-Brown identity

valid for natural numbers up to , where and are arbitrary parameters and denotes the -fold convolution of , and discussed in this previous blog post; this is the top order coefficient of

and arises by first observing that

vanishes up to , and then expanding the left-hand side using the binomial formula and the identity .

One consequence of this phenomenon is that identities involving derived multiplicative functions tend to have a dimensional consistency property: all terms in the identity have the same order of derivation in them. For instance, all the terms in the Selberg symmetry formula (3) are doubly derived functions, all the terms in the Vaughan identity (4) or the Heath-Brown identity (5) are singly derived functions, and so forth. One can then use dimensional analysis to help ensure that one has written down a key identity involving such functions correctly, much as is done in physics.

In addition to the dimensional analysis arising from the order of derivation, there is another dimensional analysis coming from the value of multiplicative functions at primes (which is more or less equivalent to the order of pole of the Dirichlet series at ). Let us say that a multiplicative function has a *pole of order * if one has on the average for primes , where we will be a bit vague as to what “on the average” means as it usually does not matter in applications. Thus for instance, or has a pole of order (a simple pole), or has a pole of order (i.e. neither a zero or a pole), Dirichlet characters also have a pole of order (although this is slightly nontrivial, requiring Dirichlet’s theorem), has a pole of order (a simple zero), has a pole of order , and so forth. Note that the convolution of a multiplicative function with a pole of order with a multiplicative function with a pole of order will be a multiplicative function with a pole of order . If there is no oscillation in the primes (e.g. if for *all* primes , rather than on the average), it is also true that the product of a multiplicative function with a pole of order with a multiplicative function with a pole of order will be a multiplicative function with a pole of order . The situation is significantly different though in the presence of oscillation; for instance, if is a quadratic character then has a pole of order even though has a pole of order .

A -derived multiplicative function will then be said to have an *underived pole of order * if it is the top order coefficient of a multiplicative function with a pole of order ; in terms of Dirichlet series, this roughly means that the Dirichlet series has a pole of order at . For instance, the singly derived multiplicative function has an underived pole of order , because it is the top order coefficient of , which has a pole of order ; similarly has an underived pole of order , being the top order coefficient of . More generally, and have underived poles of order and respectively for any .

By taking top order coefficients, we then see that the convolution of a -derived multiplicative function with underived pole of order and a -derived multiplicative function with underived pole of order is a -derived multiplicative function with underived pole of order . If there is no oscillation in the primes, the product of these functions will similarly have an underived pole of order , for instance has an underived pole of order . We then have the dimensional consistency property that in any of the standard identities involving derived multiplicative functions, all terms not only have the same derived order, but also the same underived pole order. For instance, in (3), (4), (5) all terms have underived pole order (with any Mobius function terms being counterbalanced by a matching term of or ). This gives a second way to use dimensional analysis as a consistency check. For instance, any identity that involves a linear combination of and is suspect because the underived pole orders do not match (being and respectively), even though the derived orders match (both are ).

One caveat, though: this latter dimensional consistency breaks down for identities that involve infinitely many terms, such as Linnik’s identity

In this case, one can still rewrite things in terms of multiplicative functions as

so the former dimensional consistency is still maintained.

I thank Andrew Granville, Kannan Soundararajan, and Emmanuel Kowalski for helpful conversations on these topics.

Tamar Ziegler and I have just uploaded to the arXiv our paper “Narrow progressions in the primes“, submitted to the special issue “Analytic Number Theory” in honor of the 60th birthday of Helmut Maier. The results here are vaguely reminiscent of the recent progress on bounded gaps in the primes, but use different methods.

About a decade ago, Ben Green and I showed that the primes contained arbitrarily long arithmetic progressions: given any , one could find a progression with consisting entirely of primes. In fact we showed the same statement was true if the primes were replaced by any subset of the primes of positive relative density.

A little while later, Tamar Ziegler and I obtained the following generalisation: given any and any polynomials with , one could find a “polynomial progression” with consisting entirely of primes. Furthermore, we could make this progression somewhat “narrow” by taking (where denotes a quantity that goes to zero as goes to infinity). Again, the same statement also applies if the primes were replaced by a subset of positive relative density. My previous result with Ben corresponds to the linear case .

In this paper we were able to make the progressions a bit narrower still: given any and any polynomials with , one could find a “polynomial progression” with consisting entirely of primes, and such that , where depends only on and (in fact it depends only on and the degrees of ). The result is still true if the primes are replaced by a subset of positive density , but unfortunately in our arguments we must then let depend on . However, in the linear case , we were able to make independent of (although it is still somewhat large, of the order of ).

The polylogarithmic factor is somewhat necessary: using an upper bound sieve, one can easily construct a subset of the primes of density, say, , whose arithmetic progressions of length all obey the lower bound . On the other hand, the prime tuples conjecture predicts that if one works with the actual primes rather than dense subsets of the primes, then one should have infinitely many length arithmetic progressions of bounded width for any fixed . The case of this is precisely the celebrated theorem of Yitang Zhang that was the focus of the recently concluded Polymath8 project here. The higher case is conjecturally true, but appears to be out of reach of known methods. (Using the multidimensional Selberg sieve of Maynard, one can get primes inside an interval of length , but this is such a sparse set of primes that one would not expect to find even a progression of length three within such an interval.)

The argument in the previous paper was unable to obtain a polylogarithmic bound on the width of the progressions, due to the reliance on a certain technical “correlation condition” on a certain Selberg sieve weight . This correlation condition required one to control arbitrarily long correlations of , which was not compatible with a bounded value of (particularly if one wanted to keep independent of ).

However, thanks to recent advances in this area by Conlon, Fox, and Zhao (who introduced a very nice “densification” technique), it is now possible (in principle, at least) to delete this correlation condition from the arguments. Conlon-Fox-Zhao did this for my original theorem with Ben; and in the current paper we apply the densification method to our previous argument to similarly remove the correlation condition. This method does not fully eliminate the need to control arbitrarily long correlations, but allows most of the factors in such a long correlation to be *bounded*, rather than merely controlled by an unbounded weight such as . This turns out to be significantly easier to control, although in the non-linear case we still unfortunately had to make large compared to due to a certain “clearing denominators” step arising from the complicated nature of the Gowers-type uniformity norms that we were using to control polynomial averages. We believe though that this an artefact of our method, and one should be able to prove our theorem with an that is uniform in .

Here is a simple instance of the densification trick in action. Suppose that one wishes to establish an estimate of the form

for some real-valued functions which are bounded in magnitude by a weight function , but which are not expected to be bounded; this average will naturally arise when trying to locate the pattern in a set such as the primes. Here I will be vague as to exactly what range the parameters are being averaged over. Suppose that the factor (say) has enough uniformity that one can already show a smallness bound

whenever are bounded functions. (One should think of as being like the indicator functions of “dense” sets, in contrast to which are like the normalised indicator functions of “sparse” sets). The bound (2) cannot be directly applied to control (1) because of the unbounded (or “sparse”) nature of and . However one can “densify” and as follows. Since is bounded in magnitude by , we can bound the left-hand side of (1) as

The weight function will be normalised so that , so by the Cauchy-Schwarz inequality it suffices to show that

The left-hand side expands as

Now, it turns out that after an enormous (but finite) number of applications of the Cauchy-Schwarz inequality to steadily eliminate the factors, as well as a certain “polynomial forms condition” hypothesis on , one can show that

(Because of the polynomial shifts, this requires a method known as “PET induction”, but let me skip over this point here.) In view of this estimate, we now just need to show that

Now we can reverse the previous steps. First, we collapse back to

One can bound by , which can be shown to be “bounded on average” in a suitable sense (e.g. bounded norm) via the aforementioned polynomial forms condition. Because of this and the Hölder inequality, the above estimate is equivalent to

By setting to be the signum of , this is equivalent to

This is halfway between (1) and (2); the sparsely supported function has been replaced by its “densification” , but we have not yet densified to . However, one can shift by and repeat the above arguments to achieve a similar densificiation of , at which point one has reduced (1) to (2).

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