Speaker
Ken Ono (University of Virginia)
Date
October 5, 8:30-9:30, Japan (October 4, 19:30-20:30, Virginia, USA)
Title
Variants of Lehmer's Conjecture on Ramanujan's tau-function
Abstract
In the spirit of Lehmer's unresolved speculation on the non-vanishing 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 modular form. In joint work with J. Balakrishnan, W. Craig, and W.-L. Tsai, the speaker has obtained the first results for such questions. For example, infinitely many spaces are presented for which the primes \(\ell\geq 37\) are not absolute values of coefficients of any newforms with integer coefficients. For Ramanujan's tau-function, we show that
\begin{align}
\tau(n)\not\in\{\pm\ell : \ell<100 \text{ odd prime}\}
\end{align}