Refereed journal article or data article (A1)

The Goldbach Conjecture With Summands In Arithmetic Progressions

List of Authors: Salmensuu Juho

Publisher: OXFORD UNIV PRESS

Publication year: 2022

Journal: Quarterly Journal of Mathematics

Journal name in source: QUARTERLY JOURNAL OF MATHEMATICS

Journal acronym: Q J MATH

Number of pages: 27

ISSN: 0033-5606

eISSN: 1464-3847

DOI: http://dx.doi.org/10.1093/qmath/haac008

URL: https://doi.org/10.1093/qmath/haac008

Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/175192603

Abstract

We prove that, for almost all r <= N-1/2/log(O(1)) N, for any given b(1) (mod r) with (b(1), r) = 1, and for almost all b(2) (mod r) with (b(2), r) = 1, we have that almost all natural numbers 2(n) <= N with 2n b(1) + b(2) (mod r) can be written as the sum of two prime numbers 2n = p(1) + p(2), where p(1) b(1) (mod r) and p(2) b(2) (mod r) . This improves the previous result which required r <= N-1/3/log(O(1)) N instead of r <= N-1/2/log(O(1))N. We also improve some other results concerning variations of the problem.

We prove that, for almost all r <= N-1/2/log(O(1)) N, for any given b(1) (mod r) with (b(1), r) = 1, and for almost all b(2) (mod r) with (b(2), r) = 1, we have that almost all natural numbers 2(n) <= N with 2n b(1) + b(2) (mod r) can be written as the sum of two prime numbers 2n = p(1) + p(2), where p(1) b(1) (mod r) and p(2) b(2) (mod r) . This improves the previous result which required r <= N-1/3/log(O(1)) N instead of r <= N-1/2/log(O(1))N. We also improve some other results concerning variations of the problem.

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