On the Goldbach problem with restricted primes

TL;DR

This paper establishes an asymptotic formula for representing large odd integers as sums of three primes with one prime below a certain threshold, improving bounds under different hypotheses.

Contribution

It introduces new bounds for the restricted prime Goldbach problem using zero-density estimates and GRH assumptions.

Findings

Unconditional bound: U= N^{4/49} exp(log^{2/3+ε} N)

Conditional bound under GRH: U= log^{4+ε} N

Asymptotic formula for prime representations with restricted prime size

Abstract

Let $N$ be a sufficiently large, odd integer. We prove an asymptotic formula for the number of representations of $N$ as the sum of three primes, one of which is smaller than a given $U$. By inserting the currently best zero-density estimate for Dirichlet $L$-functions, we may unconditionally take $U= N^{\frac{4}{49}}\exp(\log^{\frac{2}{3}+\varepsilon}N)$ for any $\varepsilon>0$. If we assume the Generalized Riemann Hypothesis instead, we may take $U= \log^{4+\varepsilon}N$.