Minor arcs for Goldbach's problem

TL;DR

This paper advances the proof of the ternary Goldbach conjecture by providing improved bounds on minor and major arcs, introducing new methods to optimize exponential sum estimates crucial for the conjecture.

Contribution

It introduces a general method to reduce Vaughan's identity costs and exploits minor arc tails using the large sieve, enabling progress towards the conjecture.

Findings

New bounds on minor arcs and tails of major arcs

A novel approach to reduce Vaughan's identity costs

Enhanced estimates for exponential sums in Goldbach's problem

Abstract

The ternary Goldbach conjecture states that every odd number n>=7 is the sum of three primes. The estimation of sums of the form \sum_{p\leq x} e(\alpha p), \alpha = a/q + O(1/q^2), has been a central part of the main approach to the conjecture since (Vinogradov, 1937). Previous work required q or x to be too large to make a proof of the conjecture for all n feasible. The present paper gives new bounds on minor arcs and the tails of major arcs. This is part of the author's proof of the ternary Goldbach conjecture. The new bounds are due to several qualitative improvements. In particular, this paper presents a general method for reducing the cost of Vaughan's identity, as well as a way to exploit the tails of minor arcs in the context of the large sieve.