A variant of the Bombieri-Vinogradov theorem in short intervals with applications

TL;DR

This paper extends the Bombieri-Vinogradov theorem to short intervals and non-abelian settings, enabling new results on prime distributions and modular form coefficients in short intervals with splitting conditions.

Contribution

It generalizes the classical theorem to a non-abelian, short interval context and applies it to prime distribution and modular form nonvanishing in these intervals.

Findings

Primes in short intervals with splitting conditions are densely clustered.

Dense clusters of fundamental discriminants with nonvanishing quadratic twist L-values.

New arithmetic applications related to Serre's questions on modular forms.

Abstract

We generalize the classical Bombieri-Vinogradov theorem to a short interval, non-abelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are "twisted" by a splitting condition in a Galois extension $L/K$ of number fields. Using this result in conjunction with recent work of Maynard, we prove that rational primes in short intervals with a given splitting condition in a Galois extension $L/\mathbb{Q}$ exhibit dense clusters in short intervals. We explore several arithmetic applications related to questions of Serre regarding the nonvanishing Fourier coefficients of cuspidal modular forms, including finding dense clusters of fundamental discriminants $ d $ in short intervals for which the central values of $d$-quadratic twists of modular $L$-functions are non-vanishing.