Restricted Ternary Goldbach

Overview

We prove that any sufficiently large odd integer $X$ can be written as the sum of three primes, $X = p_1 + p_2 + p_3$, where $p_1 \leq X^{\frac{21}{800}+\varepsilon}$. This strictly improves a previous bound by Y. Cai. Our approach replaces the traditional vector-sieve inequality by combining the circle method with minorants and a non-negative model. Additionally, we establish a general theorem that separates the minorant construction from the circle method machinery.

This paper was written as part of the Summer Research in Mathematics (SRIM) programme at the University of Cambridge, under the supervision of Dr Lasse Grimmelt and Dr Joni Teräväinen.