Paper Title:

Low lying zeros of Rankin-Selberg $L$-functions

Published on:

23 August 2023

Primary Category:

Number Theory

Paper Authors:

Alexander Shashkov

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Assumes GRH and computes 1-level density of Rankin-Selberg L-functions

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Averages over families of GL(2) x GL(2) Rankin-Selberg convolutions

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Shows the 1-level density matches predictions from random matrix theory

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Obtains support up to (-5/4, 5/4) by analyzing Kloosterman and Gauss sums

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Carefully handles contribution from poles when levels are equal

Low lying zeros of Rankin-Selberg L-functions

This paper studies the low-lying zeros of GL(2) x GL(2) Rankin-Selberg L-functions. It assumes GRH and computes the 1-level density averaged over families of these L-functions. The Katz-Sarnak density conjecture predicts the 1-level density matches that of the symplectic unitary ensemble from random matrix theory. The paper shows this holds for test functions supported in (-1/2, 1/2) in general, and (-5/4, 5/4) or (-29/28, 29/28) in certain cases, providing evidence for the conjecture.

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