  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.
