On q-analogues of Riemann's zeta

Abstract

In the paper, we introduce $q$-deformations of the Riemann zeta function, extend them to the whole complex plane, and establish certain estimates of the number of roots. The construction is based on the recent difference generalization of the Harish-Chandra theory of zonal spherical functions. We also discuss numerical results, which indicate that the location of the zeros of the $q$-zeta functions is far from random.