Selecta Mathematica

, Volume 7, Issue 4, pp 447-491

On q-analogues of Riemann's zeta function

  • I. CherednikAffiliated withDepartment of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, USA,¶ e-mail: chered@math.unc.edu

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Abstract.

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.

Key words. Zeta function, Q-gamma function, orthogonal polynomials.