Ramanujan's function on small primes

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.