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

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• 作者：Claus Bauer
• 关键词：Exponential sums ; Prime numbers ; Dirichlet series
• 刊名：Archiv der Mathematik
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：108
• 期：2
• 页码：159-172
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1420-8938
• 卷排序：108

文摘

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.