An Orthogonal Test of the $L$-functions Ratios Conjecture, II

TL;DR

This paper rigorously verifies the predictions of the L-functions Ratios Conjecture for the 1-level density of certain families of cuspidal newforms, confirming its accuracy up to square-root precision and exploring the significance of lower order terms.

Contribution

It provides the most comprehensive validation to date of the Ratios Conjecture's predictions for the 1-level density, including analysis of error term cancellations and non-diagonal contributions.

Findings

Confirmed the Ratios Conjecture's predictions up to square-root accuracy for support in (-1,1)

Extended validation to support in (-2,2) with power savings

Identified the role of non-diagonal terms in the conjecture's accuracy

Abstract

Recently Conrey, Farmer, and Zirnbauer developed the L-functions Ratios conjecture, which gives a recipe that predicts a wealth of statistics, from moments to spacings between adjacent zeros and values of L-functions. The problem with this method is that several of its steps involve ignoring error terms of size comparable to the main term; amazingly, the errors seem to cancel and the resulting prediction is expected to be accurate up to square-root cancellation. We prove the accuracy of the Ratios Conjecture's prediction for the 1-level density of families of cuspidal newforms of constant sign (up to square-root agreement for support in (-1,1), and up to a power savings in (-2,2)), and discuss the arithmetic significance of the lower order terms. This is the most involved test of the Ratios Conjecture's predictions to date, as it is known that the error terms dropped in some of the steps do not cancel, but rather contribute a main term! Specifically, these are the non-diagonal terms in the Petersson formula, which lead to a Bessel-Kloosterman sum which contributes only when the support of the Fourier transform of the test function exceeds (-1, 1).