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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
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.
for any (non-principal) 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.
- 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 1 If 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 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“.
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 ofr 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).
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 1 The elementary and complex-analytic approaches to multiplicative number theory are the two classical approaches to the subject. One could also consider a more “Fourier-analytic” approach, in which one studies convolution-type statistics such as
as 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.
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
for every natural number .
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.