Variations of Lehmer’s Conjecture on the Nonvanishing of the Ramanujan tau-function

Abstract: In the spirit of Lehmer's unresolved speculation on the nonvanishing of Ramanujan's tau-function, it is natural to ask whether a fixed integer is a value of $\tau(n)$, or is a Fourier coefficient of any given newform. In joint work with J. Balakrishnan, W. Craig, and W.-L. Tsai, the speaker has obtained some results that will be described here. For example, infinitely many spaces are presented for which the primes $\ell\leqslant 37$ are not absolute values of coefficients of any new forms with integer coefficients. For Ramanujan’s tau-function, such results imply, for $n>1$, that $\tau(n)\notin \{\pm \ell\,:\, \ell<100 \,\text{is odd prime}\}$.

number theory

Audience: researchers in the topic

EIMI Number Theory Seminar

Series comments: Password: the number of quadratic nonresidues modulo 23