Archiv der Mathematik

, Volume 108, Issue 2, pp 159–172 | Cite as

Goldbach’s conjecture in arithmetic progressions: number and size of exceptional prime moduli

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Abstract

Set \({T=N^{\frac{1}{3}-\epsilon}}\). It is proved that for all but \({\ll TL^{-H},\,H > 0}\), exceptional prime numbers \({k\leq T}\) and almost all integers b1, b2 co-prime to k, almost all integers \({n\sim N}\) satisfying \({n\equiv b_{1}+b_{2}(mod\,k)}\) can be written as the sum of two primes p1 and p2 satisfying \({p_{i}\equiv b_{i}(mod\,k),\,i=1,2}\). For prime numbers \({k\leq N^{\frac{5}{24}-\epsilon}}\), this result is even true for all but \({\ll (\log\,N)^{D}}\) primes k and all integers b1, b2 co-prime to k.

Keywords

Exponential sums Prime numbers Dirichlet series 

Mathematics Subject Classification

11F32 11F25 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Dolby LaboratoriesBeijingChina

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