A higher-dimensional Siegel-Walfisz theorem

TL;DR

This paper extends the Siegel-Walfisz theorem to higher dimensions, providing new asymptotic results for prime patterns in dense subsets of primes, surpassing previous one-dimensional limitations.

Contribution

It proves a higher-dimensional version of the Siegel-Walfisz theorem, enabling asymptotic counts of prime tuples in more complex settings.

Findings

The theorem applies to systems of affine-linear forms with bounded coefficients.

It counts arithmetic progressions of primes with step sizes related to log N.

First asymptotic results for primes where p-1 is squarefree in dense subsets.

Abstract

The Green-Tao-Ziegler theorem provides asymptotics for the number of prime tuples of the form $(\psi_1(n),\ldots,\psi_t(n))$ when $n$ ranges among the integer vectors of a convex body $K\subset [-N,N]^d$ and $\Psi=(\psi_1,\ldots,\psi_t)$ is a system of affine-linear forms whose linear coefficients remain bounded (in terms of $N$). In the $t=1$ case, the Siegel-Walfisz theorem shows that the asymptotic still holds when the coefficients vary like a power of $\log N$. We prove a higher-dimensional (i.e. $t>1$) version of this fact. We provide natural examples where our theorem goes beyond the one of Green and Tao, such as the count of arithmetic of progressions of step $\lfloor \log N\rfloor$ times a prime in the primes up to $N$. We also apply our theorem to the determination of asymptotics for the number of linear patterns in a dense subset of the primes, namely the primes $p$ for which $p-1$ is squarefree. To the best of our knowledge, this is the first such result in dense subsets of primes save for congruence classes.