An Orthogonal Test of the L-Functions Ratios Conjecture

TL;DR

This paper empirically tests the L-functions Ratios Conjecture for families of cuspidal newforms, confirming its predictions for 1-level density up to a certain support, and explores different sum extensions leading to varying results.

Contribution

It provides the first detailed empirical validation of the Ratios Conjecture's predictions for 1-level density in these families, including weighted and unweighted cases.

Findings

Confirmed the Ratios Conjecture predictions up to power savings for test functions supported in (-2, 2)

Showed the equivalence of weighted and unweighted 1-level densities in the tested cases

Identified alternative sum extensions leading to different lower order terms

Abstract

We test the predictions of the L-functions Ratios Conjecture for the family of cuspidal newforms of weight k and level N, with either k fixed and N --> oo through the primes or N=1 and k --> oo. We study the main and lower order terms in the 1-level density. We provide evidence for the Ratios Conjecture by computing and confirming its predictions up to a power savings in the family's cardinality, at least for test functions whose Fourier transforms are supported in (-2, 2). We do this both for the weighted and unweighted 1-level density (where in the weighted case we use the Petersson weights), thus showing that either formulation may be used. These two 1-level densities differ by a term of size 1 / log(k^2 N). Finally, we show that there is another way of extending the sums arising in the Ratios Conjecture, leading to a different answer (although the answer is such a lower order term that it is hopeless to observe which is correct).