Primes in simultaneous arithmetic progressions

TL;DR

This paper establishes a new mean value theorem for the distribution of primes in two simultaneous arithmetic progressions, combining spectral theory and algebraic geometry techniques, with applications to prime factors of Chen primes.

Contribution

It introduces a novel mean value theorem for primes in simultaneous progressions using advanced spectral and algebraic methods, extending previous analytic approaches.

Findings

Proves infinitely often the greatest prime factor of p+6 exceeds p^{0.217} for Chen primes.

Develops estimates for exponential sums from automorphic forms and algebraic geometry.

Enhances understanding of prime distribution in arithmetic progressions.

Abstract

We prove a new mean value theorem on the distribution of primes in two simultaneous arithmetic progressions. Our approach builds on previous arguments of Bombieri, Fouvry, Friedlander, and Iwaniec appealing to spectral theory of Kloosterman sums, as well as the $q$-analogue of van der Corput method. In particular, we need estimates for exponential sums coming from the spectral theory of automorphic forms (sums of Kloosterman sums) and from algebraic geometry (Weil--Deligne bound for algebraic exponential sums). As an application, we show that the greatest prime factor of $p + 6$ for Chen prime $p$ is infinitely often greater than $p^{0.217}$.