In the paper, we introduce q-deformations of the Riemann zeta function, extend them to the whole complex plane, establish a q-counterpart of the TlogT-formula for the number of zeros, and discuss theoretically and (mainly) numerically a q-variant of the Riemann hypothesis. The construction is closely related to the recent difference generalization of the Harish-Chandra theory of zonal spherical functions. The q-zeta functions do not satisfy the functional equation but have some analytic advantages.
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