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A fundamental problem in analytic number theory is to understand the distribution of the prime numbers ${\{2,3,5,\ldots\}}$. For technical reasons, it is convenient not to study the primes directly, but a proxy for the primes known as the von Mangoldt function ${\Lambda: {\mathbb N} \rightarrow {\mathbb R}}$, defined by setting ${\Lambda(n)}$ to equal ${\log p}$ when ${n}$ is a prime ${p}$ (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

$\displaystyle \log n = \sum_{d|n} \Lambda(d) \ \ \ \ \ (1)$

for every natural number ${n}$.

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 ${x}$ is equal to ${(1+o(1)) \frac{x}{\log x}}$:

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

Here, of course, ${o(1)}$ denotes a quantity that goes to zero as ${x \rightarrow \infty}$.

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

$\displaystyle \sum_{n \leq x} \Lambda(n) = (1+o(1)) x \ \ \ \ \ (2)$

for the von Mangoldt function (the key point being that the squares, cubes, etc. of primes give a negligible contribution, so ${\sum_{n \leq x} \Lambda(n)}$ is essentially the same quantity as ${\sum_{p \leq x} \log p}$). Understanding the nature of the ${o(1)}$ term is a very important problem, with the conjectured optimal decay rate of ${O(\sqrt{x} \log x)}$ 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 ${a}$ and ${q}$ with ${a}$ coprime to ${q}$ (thus ${(a,q)=1}$), the number of primes less than ${x}$ equal to ${a}$ mod ${q}$ is equal to ${(1+o_q(1)) \frac{1}{\phi(q)} \frac{x}{\log x}}$, where ${\phi(q) := \# \{ 1 \leq a \leq q: (a,q)=1 \}}$ is the Euler totient function:

$\displaystyle \sum_{p \leq x: p = a \hbox{ mod } q} 1 = (1+o_q(1)) \frac{1}{\phi(q)} \frac{x}{\log x}.$

(Of course, if ${a}$ is not coprime to ${q}$, the number of primes less than ${x}$ equal to ${a}$ mod ${q}$ is ${O(1)}$. The subscript ${q}$ in the ${o()}$ and ${O()}$ notation denotes that the implied constants in that notation is allowed to depend on ${q}$.) This is a more quantitative version of Dirichlet’s theorem, which asserts the weaker statement that the number of primes equal to ${a}$ mod ${q}$ 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 ${2}$ mod ${3}$, rather than ${1}$ mod ${3}$. Then almost all sums of three odd primes would be divisible by ${3}$, leaving dangerously few sums left to cover the remaining two residue classes. Similarly for other moduli than ${3}$. 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

$\displaystyle \sum_{n \leq x: n = a \hbox{ mod } q} \Lambda(n) = (1+o_q(1)) \frac{1}{\phi(q)} x.$

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 ${n}$ is prime is not asymptotic to ${\frac{1}{\log n}}$ (as is given by the prime number theorem), but instead to fluctuate depending on the phase of the complex number ${n^{it}}$ for some fixed real number ${t}$, thus for instance the probability might be significantly less than ${1/\log n}$ when ${t \log n}$ is close to an integer, and significantly more than ${1/\log n}$ when ${t \log n}$ 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 ${1+it}$.

In the above scenario, the primality of a large integer ${n}$ was somehow sensitive to asymptotic or “Archimedean” information about ${n}$, namely the approximate value of its logarithm. In modern terminology, this information reflects the local behaviour of ${n}$ at the infinite place ${\infty}$. There are also potential consipracies in which the primality of ${n}$ is sensitive to the local behaviour of ${n}$ at finite places, and in particular to the residue class of ${n}$ mod ${q}$ for some fixed modulus ${q}$. For instance, given a Dirichlet character ${\chi: {\mathbb Z} \rightarrow {\mathbb C}}$ of modulus ${q}$, i.e. a completely multiplicative function on the integers which is periodic of period ${q}$ (and vanishes on those integers not coprime to ${q}$), one could imagine a scenario in which the probability that a randomly chosen large integer ${n}$ is prime is large when ${\chi(n)}$ is close to ${+1}$, and small when ${\chi(n)}$ is close to ${-1}$, which would contradict the prime number theorem in arithmetic progressions. (Note the similarity between this scenario at ${q}$ and the previous scenario at ${\infty}$; in particular, observe that the functions ${n \rightarrow \chi(n)}$ and ${n \rightarrow n^{it}}$ are both totally multiplicative.) In the language of Dirichlet series, this conspiracy is more commonly known as a zero of the ${L}$-function of ${\chi}$ at ${1}$.

An especially difficult scenario to eliminate is that of real characters, such as the Kronecker symbol ${\chi(n) = \left( \frac{n}{q} \right)}$, in which numbers ${n}$ which are quadratic nonresidues mod ${q}$ are very likely to be prime, and quadratic residues mod ${q}$ 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 ${1}$, or order ${0}$ (i.e. are not zeroes after all), but cannot have an intermediate order between ${0}$ and ${1}$.) 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, ${n^{it}}$, then they must also correlate with ${n^{-it}}$. (In the language of Dirichlet series, this reflects the fact that the zeta function and ${L}$-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 ${n^{it}}$ for any non-zero ${t}$, 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 ${\chi(n)}$, then they would also correlate with the conjugate ${\overline{\chi}(n)}$, which also is inconsistent with the all-or-nothing dichotomy, except in the exceptional case when ${\chi}$ is real – which essentially means that ${\chi}$ 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 ${\infty}$ 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.