A higher-dimensional Siegel–Walfisz theorem

Pierre-Yves Bienvenu¹

¹ School of Mathematics University of Bristol Bristol BS8 1TW, United Kingdom

Acta Arithmetica, Tome 179 (2017) no. 1, pp. 79-100

Résumé

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.

Détail

DOI : 10.4064/aa8600-10-2016

Mots-clés : green tao ziegler theorem provides asymptotics number prime tuples form psi ldots psi ranges integer vectors convex body subset n varpsi psi ldots psi system affine linear forms whose linear coefficients remain bounded terms nbsp siegel walfisz theorem shows asymptotic still holds coefficients vary power log prove higher dimensional version provide natural examples where theorem goes beyond green tao count arithmetic progressions step lfloor log rfloor times prime primes apply theorem determination asymptotics number linear patterns dense subset primes namely primes which p squarefree best knowledge first result dense subsets primes save congruence classes