TY - JOUR

T1 - A higher-dimensional Siegel-Walfisz theorem

AU - Bienvenu, Pierre Yves

PY - 2017/4/28

Y1 - 2017/4/28

N2 - The Green-Tao-Ziegler theorem provides asymptotics for the number of prime tuples of the form (ψ1(n),…,ψt(n)) when n ranges among the integer vectors of a convex body K⊂[−N,N]d and Ψ=(ψ1,…,ψt) 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 logN. 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 ⌊logN⌋ 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.

AB - The Green-Tao-Ziegler theorem provides asymptotics for the number of prime tuples of the form (ψ1(n),…,ψt(n)) when n ranges among the integer vectors of a convex body K⊂[−N,N]d and Ψ=(ψ1,…,ψt) 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 logN. 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 ⌊logN⌋ 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.

KW - Green-Tao theorem

KW - Prime tuples

UR - http://www.scopus.com/inward/record.url?scp=85020302666&partnerID=8YFLogxK

U2 - 10.4064/aa8600-10-2016

DO - 10.4064/aa8600-10-2016

M3 - Article (Academic Journal)

AN - SCOPUS:85020302666

VL - 179

SP - 79

EP - 100

JO - Acta Arithmetica

JF - Acta Arithmetica

SN - 0065-1036

IS - 1

ER -