Selecta Mathematica

, Volume 7, Issue 4, pp 447–491

On q-analogues of Riemann's zeta function

  • I. Cherednik

DOI: 10.1007/s00029-001-8095-6

Cite this article as:
Cherednik, I. Sel. math., New ser. (2001) 7: 447. doi:10.1007/s00029-001-8095-6


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. 

Copyright information

© Birkhäuser Verlag, Basel, 2001

Authors and Affiliations

  • I. Cherednik
    • 1
  1. 1.Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, USA,¶ e-mail: chered@math.unc.eduUS

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