The Bombieri-Vinogradov theorem for nilsequences

The Bombieri-Vinogradov theorem for nilsequences, Discrete Analysis 2021:21, 55 pp.

The prime number theorem asserts that the density of the primes in the vicinity of a large integer $n$ is approximately $1/\log n$, or equivalently that the number of primes up to $n$ is approximately equal to the logarithmic integral $\int_0^n(\log t)^{-1}\,dt$ (which in turn is close to $n/\log n$ but when one is discussing the size of the error it is important to use the slightly more complicated expression).

Dirichlet’s theorem asserts that every infinite arithmetic progression $a,a+q,a+2q,\dots$ such that $a$ and $q$ are coprime contains infinitely many primes. It is natural to ask whether there is a simultaneous generalization of this result and the prime number theorem, and indeed there is: the Siegel-Walfisz theorem gives an estimate for the number of primes up to $n$ in a given arithmetic progression, and shows that when $n$ is large, the primes are roughly equidistributed among the allowable residue classes. That is, if $a$ and $q$ are coprime, then the proportion of primes congruent to $a$ mod $q$ converges to $1/\phi(q)$, where $\phi$ is Euler’s totient function.

A convenient way to express these results is by using the von Mangoldt function $\Lambda$, which is defined on the positive integers by taking $\Lambda(n)$ to be $\log p$ if $n=p^k$ for some prime $p$, and 0 otherwise. One can regard $\Lambda$ as correcting for the density of the primes: that is, because it takes the value $\log p$ at a prime $p$, the average size of $\Lambda$ near a large integer $n$ is roughly 1. (This does not explain why one takes $\Lambda$ to be non-zero at prime powers: the reason for choosing this definition is that it has a negligible effect on the average value of $\Lambda$, and for every positive integer $n$ one has that $\sum_{d|n}\Lambda(d)=\log n$, as can easily be checked.) The prime number theorem is then equivalent to the statement that $\sum_{n\leq x}\Lambda(n)$ is approximately $x$ for large $x$, and the Siegel-Walfisz theorem states that $\sum_{n\leq x, n\equiv a\ \text{mod}\ q}\Lambda(n)$ is approximately $x/\phi(q)$. More precisely, it states this with an error term of size roughly $x\exp(-C\sqrt{\log x})$, where $C$ depends on the size of $\log q/\log\log x$, so if $q$ is bounded by some fixed power of $\log x$, then $C$ is bounded (but the proof of the theorem is ineffective, so no estimate for $C$ is known – this is because of the possible existence of a so-called Siegel zero, which is a zero of a Dirichlet L-function that is very close to 1).

The generalized Riemann hypothesis implies a much better estimate for this error term: it brings it down to $x^{1/2+\epsilon}$ for any $\epsilon>0$. But in the absence of a spectacular breakthrough, we have to content ourselves with the weaker estimate coming from the Siegel-Walfisz theorem if we want unconditional results.

However, it turns out that for many applications to statements about the primes, one does not in fact need to know accurate estimates for how they are distributed in all arithmetic progressions – it is sufficient if one has good estimates for almost all arithmetic progressions in a suitable sense. The Bombieri-Vinogradov theorem is a major result in analytic number theory that shows that on average the error term is of the kind of size that the generalized Riemann hypothesis predicts.

One can understand the statement of the theorem as follows. For any given $x$ and $q$, let us measure the error $E(q)$ associated with the common difference $q$ as
$$\max_{y\leq x}\max_{(a,q)=1)}|\sum_{n\leq y,\ n\equiv a\text{ mod }q}\Lambda(n)-y/\phi(q)|.$$
In other words, it is the worst error that can be obtained on an arithmetic progression of integers less than or equal to $x$ that belong to a residue class mod $q$. The Bombieri-Vinogradov theorem concerns the average value of $E(q)$ over a suitable range. If $x^{1/2}(\log x)^{-A}\leq Q\leq x^{1/2}$ for some constant $A$, then it states that the average of $E(q)$ over all $q\leq Q$ is at most $x^{1/2}(\log x)^5$.

Proving results about the behaviour of primes in arithmetic progressions is closely related to proving results about exponential sums of the form $\sum_{n\leq x}\Lambda(n)e(\alpha n)$. (Here $e(x)$ stands for $e^{2\pi i x}$.) This paper considers more general sums, such as sums of the form $\sum_{n\leq x}\Lambda(n)e(P(n))$, where $P$ is a polynomial, or more generally still sums of the form $\sum_{n\leq x}\Lambda(n)\psi(n)$ where $\psi(n)$ is an object known as a nilsequence. Nilsequences are generalizations of polynomial phase functions $e(P(n))$ that play a central role in “higher-order Fourier analysis”, where they replace the role of the trigonometric functions $e(\alpha n)$ and allow one to obtain results about more general configurations. In particular, they were crucial to results of Green, Tao and Ziegler concerning solutions of linear equations in the primes, which generalized their famous theorem that the primes contain arbitrarily long arithmetic progressions.

The paper obtains Bombieri-Vinogradov-type results for these more general sums, which allows them to obtain a remarkably diverse collection of applications. One of these is a strengthening of the Green-Tao theorem, which states that if $Q\leq x^{1/4-\epsilon}$, then for almost all $q\leq Q$, every residue class $\{n:n\equiv a\text{ mod }q\}$ with $(a,q)=1$ contains an arithmetic progression of length $k$ consisting of primes. Another is a strengthening of Zhang’s bounded-gaps theorem: they show that for any $\eta>0$ and any polynomial $P$ with at least one irrational coefficient that is not the coefficient of the constant term, the set of primes $p$ such that $P(p)$ is within $\eta$ of an integer has bounded gaps. A third is a strengthening of Chen’s theorem that there are infinitely many primes $p$ such that $p+2$ is a product of two primes. They prove that there is some $\theta>0$ such that this is true even if one insists that $P(p)$ is within $p^{-\theta}$ of an integer, where $P$ is as in the previous application. (Here $\theta$ depends on the degree of $P$.) Further applications that are slightly more technical to state can be found in the paper.