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arXiv · 1410.7765

Gaps between zeros of GL(2) $L$-functions

Owen Barrett·Brian McDonald·Steven J. Miller·Patrick Ryan·Caroline L. Turnage-Butterbaugh·Karl Winsor
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Abstract

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.

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Owen Barrett, Brian McDonald, Steven J. Miller, Patrick Ryan, Caroline L. Turnage-Butterbaugh, Karl Winsor. 2014-10-28. Gaps between zeros of GL(2) $L$-functions. https://doi.org/10.1016/j.jmaa.2015.04.007

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