One-level density for holomorphic cusp forms of arbitrary level

Owen Barrett,Paula Burkhardt,Steven J Miller,Robert Dorward,Jonathan Dewitt

doi:10.1007/s40993-017-0091-9

Abstract

In 2000 Iwaniec, Luo, and Sarnak proved for certain families of L-functions associated to holomorphic newforms of square-free level that, under the Generalized Riemann Hypothesis, as the conductors tend to infinity the one-level density of their zeros matches the one-level density of eigenvalues of large random matrices from certain classical compact groups in the appropriate scaling limit. We remove the square-free restriction by obtaining a trace formula for arbitrary level by using a basis developed by Blomer and Milićević, which is of use for other problems as well.

Highlights

1 Introduction Montgomery [1] conjectured that the pair correlation of critical zeros up to height T of the Riemann zeta function ζ (s) coincides with the pair correlation of eigenvalues of random unitary matrices of dimension N in the appropriate limit as T, N → ∞
While additional support for this agreement was obtained by the work of Hejhal [2] on the triple correlation of ζ (s), Rudnick and Sarnak [3] on the n-level correlation for cuspidal automorphic forms, and Odlyzko [4,5] on the spacings between adjacent zeros of ζ (s), the story cannot end here as these statistics are insensitive to the behavior of any finite set of zeros
Note that in many cases, the right-hand side of (1.6) is preferable to the left-hand side, as it is amenable to application of spectral summation formulas such as the Petersson formula (Proposition 2.1) and can be studied via Kloosterman sums, see Proposition 5.2

Summary

Introduction

Montgomery [1] conjectured that the pair correlation of critical zeros up to height T of the Riemann zeta function ζ (s) coincides with the pair correlation of eigenvalues of random unitary matrices of dimension N in the appropriate limit as T, N → ∞. We concentrate on extending the results of Iwaniec, Luo, and Sarnak in [18] One of their key results is a formula for unweighted sums of Fourier coefficients of holomorphic newforms of a given weight and level. In 2011, Rouymi [40] complemented the square-free calculations of Iwaniec, Luo, and Sarnak, finding an orthonormal basis for the space of cusp forms of prime power level, and applying this explicit basis towards the development of a similar sum of Fourier coefficients over all newforms with level equal to a fixed prime power. In 2015, Blomer and Milicevic [41] extended the results of Iwaniec, Luo, and Sarnak and Rouymi by writing down an explicit orthonormal basis for the space of cusp forms (holomorphic or Maass) of a fixed weight and, novelly, arbitrary level. We use this formula to show the 1-level density agrees only with orthogonal symmetry

Harmonic averaging

The Density Conjecture

Estimating tails of pure sums