2014/10/31 by Owen Barrett, Brian McDonald, Steven J. Miller +5 · 3 citations
Computer Science · Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #Coding theory and cryptography #Combinatorics #Cusp (singularity) #Function (biology) #Geometry #Holomorphic function #Mathematical analysis #Mathematics #math.NT #msc:11M26 #msc:11M41
paper · pdf · doi:10.1016/j.jmaa.2015.04.007
published in Journal of Mathematical Analysis and Applications 429(1), 204-232 (Elsevier BV) · This 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
arxiv created 2015/04/09 · openalex publication_date 2015/04/09 · arxiv updated 2015/04/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
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.