Number Theory Seminar

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 τ(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 l<=37 are not absolute values of coefficients of any newforms with integer coefficients. For Ramanujan's tau-function, we offer several infinite families of integers that are never values.