A higher-dimensional Siegel–Walfisz theorem
Tom 179 / 2017
Streszczenie
The Green–Tao–Ziegler theorem provides asymptotics for the number of prime tuples of the form 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.