Extending support for the centered moments of the low lying zeroes of cuspidal newforms

Abstract

We study low-lying zeroes of L-functions and their n-level density, which relies on a smooth test function φ whose Fourier transform φ has compact support. Assuming the generalized Riemann hypothesis, we compute the n th centered moments of the 1-level density of low-lying zeroes of L-functions associated with weight k, prime level N cuspidal newforms as N → ∞, where supp( φ) ⊂ (−2/n, 2/n). The Katz-Sarnak density conjecture predicts that the n-level density of certain families of L-functions is the same as the distribution of eigenvalues of corresponding families of orthogonal random matrices. We prove that the Katz-Sarnak density conjecture holds for the n th centered moment of the 1-level density for test functions with φ supported in (−2/n, 2/n), for families of cuspidal newforms split by the sign of their functional equations. Our work provides better bounds on the percent of forms vanishing to a certain order at the central point. Previous work handled the 1-level for support up to 2 and the n-level up to min(2/n, 1/(n − 1)); we are able to remove the second restriction on the support and extend the result to what one would expect, based on the 1-level, by finding a tractable vantage to evaluate the combinatorial zoo of terms which emerge.