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

Abstract

Let $L(s,f)$ be an $L$-function associated to a primitive (holomorphic or Maass) cusp form $f$ on GL(2) over $\mathbb{Q}$. 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 $\sqrt{3}=\mathrm{1.732...}$ 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.

arXiv commentThis article is a product of the 2014 SMALL REU at Williams College. Version 2 Comments: 25 pages, typos fixed, and references updated. We thank Micah Milinovich for his feedback. Version 3 Comment: to appear in the Journal of Mathematical Analysis and Applications. Version 4 Comments: typo in equation (1.6) fixed, in press at the Journal of Mathematical Analysis and Applications

References in corpus (8)

• The subconvexity problem for $\GL_{2}$
• On the Selberg class of Dirichlet series: small degrees
• Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function
• A note on the gaps between consecutive zeros of the Riemann zeta-function
• New bounds for automorphic L-functions
• Large gaps between consecutive zeros, on the critical line, of the Riemann zeta-function
• Simple zeros of modular L-functions
• Gaps between zeros of Dedekind zeta-functions of quadratic number fields. II

Cited by in corpus (2)

• Gaps between zeros of Dedekind zeta-functions of quadratic number fields. II
• Gaps between zeros of the Riemann zeta-function