The Goldbach Conjecture With Summands In Arithmetic Progressions

Author:

Salmensuu Juho1

Affiliation:

1. Department of Mathematics and Statistics, University of Turku , Turku, Finland

Abstract

Abstract We prove that, for almost all $r \leq N^{1/2}/\log^{O(1)}N$, for any given $b_1 \ (\mathrm{mod}\ r)$ with $(b_1, r) = 1$, and for almost all $b_2 \ (\mathrm{mod}\ r)$ with $(b_2, r) = 1$, we have that almost all natural numbers $2n \leq N$ with $2n \equiv b_1 + b_2 \ (\mathrm{mod}\ r)$ can be written as the sum of two prime numbers $2n = p_1 + p_2$, where $p_1 \equiv b_1 \ (\mathrm{mod}\ r)$ and $p_2 \equiv b_2 \ (\mathrm{mod}\ r)$. This improves the previous result which required $r \leq N^{1/3}/\log^{O(1)}N$ instead of $r \leq N^{1/2}/\log^{O(1)}N$. We also improve some other results concerning variations of the problem.

Funder

Emil Aaltonen foundation

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

Reference9 articles.

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