Article ID Journal Published Year Pages File Type
4615281 Journal of Mathematical Analysis and Applications 2015 29 Pages PDF
Abstract

Let L(s,f)L(s,f) be an L-function associated to a primitive (holomorphic or Maass) cusp form f   on GL(2)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)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)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)GL(2) over Q, we prove that there are infinitely many gaps between consecutive zeros of L(s,f)L(s,f) along the critical line that are at most 0.823 times the average spacing.

Related Topics
Physical Sciences and Engineering Mathematics Analysis
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