Gaps between zeros of [formula omitted]L-functions

Abstract

Let L(s,f) be an L-function associated to a primitive (holomorphic or Maass) cusp form f on GL(2) over 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 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 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.