Vinogradov's three primes theorem with primes having given primitive roots

TL;DR

This paper combines Hooley's method with the Hardy-Littlewood circle method to study representations of odd integers as sums of three primes with specified primitive roots, analyzing the singular series and its Euler product structure.

Contribution

It introduces a novel approach by integrating Hooley's method with the circle method to study primes with prescribed primitive roots in additive problems.

Findings

Partial Euler product factorisation of the singular series

Identification of limitations in extending the Euler product

Insights into the distribution of primes with specific primitive roots

Abstract

The first purpose of our paper is to show how Hooley's celebrated method leading to his conditional proof of the Artin conjecture on primitive roots can be combined with the Hardy-Littlewood circle method. We do so by studying the number of representations of an odd integer as a sum of three primes, all of which have prescribed primitive roots. The second purpose is to analyse the singular series. In particular, using results of Lenstra, Stevenhagen and Moree, we provide a partial factorisation as an Euler product and prove that this does not extend to a complete factorisation.