Ramanujan's function on small primes

TL;DR

This paper empirically investigates the eigenvalues of determinants related to Ramanujan functions derived from cusp forms, exploring their oscillatory behavior and potential implications for Lehmer's conjecture.

Contribution

It introduces an empirical analysis of Ramanujan functions' eigenvalues and examines their oscillations to inform approaches to Lehmer's question.

Findings

Eigenvalues exhibit oscillatory patterns as n increases

Potential regularities in eigenvalue distributions are identified

Implications for understanding zeros of Ramanujan's tau function are discussed

Abstract

We denote functions mapping n to the Fourier coefficient of q^n in the expansion of a cusp form as Ramanujan functions. We empirically study the eigenvalues of determinants that represent values of these Ramanujan functions. In some cases, considered as point sets in the complex plane, they appear to oscillate as n increases. We look for regularities in this phenomenon and discuss the possibility of exploiting it to attack Lehmer's question about the existence of zeros of Ramanujan's tau function.