Professor Peter Sin
Department of Mathematics, University of Florida

Group actions on Dirichlet Series

Abstract:

This talk describes joint work with John G. Thompson. A Dirichlet series is an infinite series of the form $\sum_n a_n n^{-s}$. The simplest and most famous Dirichlet series is the Riemann zeta function, in which all $a_n$ are equal to 1, which converges in the complex half-plane Real(s)>1. The set of Dirichlet Series which converge somewhere forms a vector space, in fact a commutative algebra D{s} under the operation of Dirichlet convolution. The matrix representing the linear map given by multiplication by the Riemann zeta function is the divisor matrix D=(d_{ij}), where d_ij=1 if i divides j, and zero otherwise. We consider group actions on D{s} in which the group contains the matrix D. We construct an action of the group SL(2,Z) in which D represents the standard unipotent element. It is then shown that all Dirichlet series in the orbit of the zeta function are algebraically related and a cubic relation in two variables (elliptic curve) is given relating zeta with an associate (in D{s}) of (zeta-1). This observation contrasts with the classical fact that the zeta function does not satisfy any algebraic differential-difference equation.