A higher-dimensional Siegel-Walfisz theorem

Pierre Yves Bienvenu*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

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Abstract

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.
Original languageEnglish
Pages (from-to)79-100
Number of pages22
JournalActa Arithmetica
Volume179
Issue number1
DOIs
Publication statusPublished - 28 Apr 2017

Keywords

  • Green-Tao theorem
  • Prime tuples

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