A higher-dimensional Siegel–Walfisz theorem
Volume 179 / 2017
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 over the integer vectors of a convex body $K\subset [-N,N]^d$ and $\varPsi=(\psi_1,\ldots,\psi_t)$ 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. $t \gt 1$) 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 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.