You are currently browsing the tag archive for the ‘Dirichlet characters’ tag.

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 3 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.

A fundamental problem in analytic number theory is to understand the distribution of the prime numbers . For technical reasons, it is convenient not to study the primes directly, but a proxy for the primes known as the von Mangoldt function , defined by setting to equal when is a prime (or a power of that prime) and zero otherwise. The basic reason why the von Mangoldt function is useful is that it encodes the fundamental theorem of arithmetic (which in turn can be viewed as the defining property of the primes) very neatly via the identity

The most important result in this subject is the prime number theorem, which asserts that the number of prime numbers less than a large number is equal to :

Here, of course, denotes a quantity that goes to zero as .

It is not hard to see (e.g. by summation by parts) that this is equivalent to the asymptotic

for the von Mangoldt function (the key point being that the squares, cubes, etc. of primes give a negligible contribution, so is essentially the same quantity as ). Understanding the nature of the term is a very important problem, with the conjectured optimal decay rate of being equivalent to the Riemann hypothesis, but this will not be our concern here.

The prime number theorem has several important generalisations (for instance, there are analogues for other number fields such as the Chebotarev density theorem). One of the more elementary such generalisations is the prime number theorem in arithmetic progressions, which asserts that for fixed and with coprime to (thus ), the number of primes less than equal to mod is equal to , where is the Euler totient function:

(Of course, if is not coprime to , the number of primes less than equal to mod is . The subscript in the and notation denotes that the implied constants in that notation is allowed to depend on .) This is a more quantitative version of Dirichlet’s theorem, which asserts the weaker statement that the number of primes equal to mod is infinite. This theorem is important in many applications in analytic number theory, for instance in Vinogradov’s theorem that every sufficiently large odd number is the sum of three odd primes. (Imagine for instance if almost all of the primes were clustered in the residue class mod , rather than mod . Then almost all sums of three odd primes would be divisible by , leaving dangerously few sums left to cover the remaining two residue classes. Similarly for other moduli than . This does not fully rule out the possibility that Vinogradov’s theorem could still be true, but it does indicate why the prime number theorem in arithmetic progressions is a relevant tool in the proof of that theorem.)

As before, one can rewrite the prime number theorem in arithmetic progressions in terms of the von Mangoldt function as the equivalent form

Philosophically, one of the main reasons why it is so hard to control the distribution of the primes is that we do not currently have too many tools with which one can rule out “conspiracies” between the primes, in which the primes (or the von Mangoldt function) decide to correlate with some structured object (and in particular, with a totally multiplicative function) which then visibly distorts the distribution of the primes. For instance, one could imagine a scenario in which the probability that a randomly chosen large integer is prime is not asymptotic to (as is given by the prime number theorem), but instead to fluctuate depending on the phase of the complex number for some fixed real number , thus for instance the probability might be significantly less than when is close to an integer, and significantly more than when is close to a half-integer. This would contradict the prime number theorem, and so this scenario would have to be somehow eradicated in the course of proving that theorem. In the language of Dirichlet series, this conspiracy is more commonly known as a zero of the Riemann zeta function at .

In the above scenario, the primality of a large integer was somehow sensitive to asymptotic or “Archimedean” information about , namely the approximate value of its logarithm. In modern terminology, this information reflects the local behaviour of at the infinite place . There are also potential consipracies in which the primality of is sensitive to the local behaviour of at finite places, and in particular to the residue class of mod for some fixed modulus . For instance, given a Dirichlet character of modulus , i.e. a completely multiplicative function on the integers which is periodic of period (and vanishes on those integers not coprime to ), one could imagine a scenario in which the probability that a randomly chosen large integer is prime is large when is close to , and small when is close to , which would contradict the prime number theorem in arithmetic progressions. (Note the similarity between this scenario at and the previous scenario at ; in particular, observe that the functions and are both totally multiplicative.) In the language of Dirichlet series, this conspiracy is more commonly known as a zero of the -function of at .

An especially difficult scenario to eliminate is that of *real characters*, such as the Kronecker symbol , in which numbers which are quadratic nonresidues mod are very likely to be prime, and quadratic residues mod are unlikely to be prime. Indeed, there is a scenario of this form – the Siegel zero scenario – which we are still not able to eradicate (without assuming powerful conjectures such as GRH), though fortunately Siegel zeroes are not quite strong enough to destroy the prime number theorem in arithmetic progressions.

It is difficult to prove that no conspiracy between the primes exist. However, it is not entirely impossible, because we have been able to exploit two important phenomena. The first is that there is often a “all or nothing dichotomy” (somewhat resembling the *zero-one laws* in probability) regarding conspiracies: in the asymptotic limit, the primes can either conspire totally (or more precisely, anti-conspire totally) with a multiplicative function, or fail to conspire at all, but there is no middle ground. (In the language of Dirichlet series, this is reflected in the fact that zeroes of a meromorphic function can have order , or order (i.e. are not zeroes after all), but cannot have an intermediate order between and .) As a corollary of this fact, the prime numbers cannot conspire with two distinct multiplicative functions at once (by having a partial correlation with one and another partial correlation with another); thus one can use the existence of one conspiracy to exclude all the others. In other words, there is at most one conspiracy that can significantly distort the distribution of the primes. Unfortunately, this argument is *ineffective*, because it doesn’t give any control at all on what that conspiracy is, or even if it exists in the first place!

But now one can use the second important phenomenon, which is that because of symmetries, one type of conspiracy can lead to another. For instance, because the von Mangoldt function is real-valued rather than complex-valued, we have conjugation symmetry; if the primes correlate with, say, , then they must also correlate with . (In the language of Dirichlet series, this reflects the fact that the zeta function and -functions enjoy symmetries with respect to reflection across the real axis (i.e. complex conjugation).) Combining this observation with the all-or-nothing dichotomy, we conclude that the primes cannot correlate with for any non-zero , which in fact leads directly to the prime number theorem (2), as we shall discuss below. Similarly, if the primes correlated with a Dirichlet character , then they would also correlate with the conjugate , which also is inconsistent with the all-or-nothing dichotomy, except in the exceptional case when is real – which essentially means that is a quadratic character. In this one case (which is the only scenario which comes close to threatening the truth of the prime number theorem in arithmetic progressions), the above tricks fail and one has to instead exploit the algebraic number theory properties of these characters instead, which has so far led to weaker results than in the non-real case.

As mentioned previously in passing, these phenomena are usually presented using the language of Dirichlet series and complex analysis. This is a very slick and powerful way to do things, but I would like here to present the elementary approach to the same topics, which is slightly weaker but which I find to also be very instructive. (However, I will not be *too* dogmatic about keeping things elementary, if this comes at the expense of obscuring the key ideas; in particular, I will rely on multiplicative Fourier analysis (both at and at finite places) as a substitute for complex analysis in order to expedite various parts of the argument. Also, the emphasis here will be more on heuristics and intuition than on rigour.)

The material here is closely related to the theory of *pretentious characters* developed by Granville and Soundararajan, as well as an earlier paper of Granville on elementary proofs of the prime number theorem in arithmetic progressions.

## Recent Comments