%0 Journal Article
%T Gaps between zeros of GL(2) $L$-functions
%A Owen Barrett
%A Brian McDonald
%A Steven J. Miller
%A Patrick Ryan
%A Caroline L. Turnage-Butterbaugh
%A Karl Winsor
%J Mathematics
%D 2014
%I arXiv
%R 10.1016/j.jmaa.2015.04.007
%X Let $L(s,f)$ be an $L$-function associated to a primitive (holomorphic or Maass) cusp form $f$ on GL(2) over $\mathbb{Q}$. Combining mean-value estimates of Montgomery and Vaughan with a method of Ramachandra, we prove a formula for the mixed second moments of derivatives of $L(1/2+it,f)$ and, via a method of Hall, use it to show that there are infinitely many gaps between consecutive zeros of $L(s,f)$ along the critical line that are at least $\sqrt 3 = 1.732...$ times the average spacing. Using general pair correlation results due to Murty and Perelli in conjunction with a technique of Montgomery, we also prove the existence of small gaps between zeros of any primitive $L$-function of the Selberg class. In particular, when $f$ is a primitive holomorphic cusp form on GL(2) over $\mathbb{Q}$, we prove that there are infinitely many gaps between consecutive zeros of $L(s,f)$ along the critical line that are at most $< 0.823$ times the average spacing.
%U http://arxiv.org/abs/1410.7765v4