Goldbach's problem with primes in arithmetic progressions and in short intervals

TL;DR

This paper advances understanding of Goldbach's problem by establishing mean value theorems for primes in arithmetic progressions and short intervals, leading to new results on representing large odd numbers as sums of primes with specific properties.

Contribution

It introduces new mean value theorems related to primes in short intervals and arithmetic progressions, providing nontrivial estimates and applications to Goldbach's ternary problem.

Findings

Goldbach's ternary problem is solvable with primes in short intervals under certain conditions.

Primes p_1,p_2 can be chosen with p_i in [X_i,X_i+Y] and (p_1+2)(p_2+2) having at most 9 prime factors.

Established bounds on parameters , for representing large odd integers as sums of primes.

Abstract

Some mean value theorems in the style of Bombieri-Vinogradov's theorem are discussed. They concern binary and ternary additive problems with primes in arithmetic progressions and short intervals. Nontrivial estimates for some of these mean values are given. As application inter alia, we show that for large odd n\not\equiv 1 (6), Goldbach's ternary problem n=p_1+p_2+p_3 is solvable with primes p_1,p_2 in short intervals p_i \in [X_i,X_i+Y] with X_{i}^{\theta_i}=Y, i=1,2, and \theta_1,\theta_2\geq 0.933 such that (p_1+2)(p_2+2) has at most 9 prime factors.