A Variant of the Bombieri-Vinogradov Theorem for Short Intervals With Applications

Jesse Thorner

Published 2014-05-21, updated 2015-05-14Version 4

We prove a Chebotarev analogue of the Bombieri-Vinogradov theorem for short intervals. Using this result in conjunction with recent work of Maynard, we prove that Chebotarev primes (determined by a Galois extension $L/\mathbb{Q}$) exhibit dense clusters in short intervals. We explore several arithmetic applications related to a question of Serre regarding the Fourier coefficients of cuspidal modular forms. These applications include 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, or for which $d$-quadratic twists of elliptic curves over $\mathbb{Q}$ have rank zero.