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arXiv · 2207.08805

The Exceptional Set in Goldbach's Problem with Almost Twin Primes

Lasse Grimmelt·Joni Teräväinen
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Abstract

We consider the exceptional set in the binary Goldbach problem for sums of two almost twin primes. Our main result is a power-saving bound for the exceptional set in the problem of representing $m=p_1+p_2$ where $p_1+2$ has at most $2$ prime divisors and $p_2+2$ has at most $3$ prime divisors. There are three main ingredients in the proof: a new transference principle like approach for sieves, a combination of the level of distribution estimates of Bombieri--Friedlander--Iwaniec and Maynard with ideas of Drappeau to produce power savings, and a generalisation of the circle method arguments of Montgomery and Vaughan that incorporates sieve weights.

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Lasse Grimmelt, Joni Teräväinen. 2022-07-06. The Exceptional Set in Goldbach's Problem with Almost Twin Primes. https://arxiv.org/abs/2207.08805

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