On the Goldbach problem with restricted primes

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 $U=N^{\frac{4}{49}}\exp ({\log }^{\frac{2}{3}+\epsilon }N)$ for any $\epsilon >0$. If we assume the Generalized Riemann Hypothesis instead, we may take $U={\log }^{4+\epsilon }N$.