Exploring the q-Riemann Zeta function and q-Bernoulli polynomials

T. Kim, C. S. Ryoo, L. C. Jang and S. H. Rim
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 11, 2005, Number 1, Pages 7—19
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Authors and affiliations

T. Kim
Institute of Science Education
Kongju National University, Kongju 314-701, Korea

C. S. Ryoo
Department of Mathematics,
Hannam University, Daejeon 306-791, Korea

L. C. Jang
Department of Mathematics and Compute Science,
KonKuk University, Choongju 380-701, Korea

S. H. Rim
Department of Mathematics Education,
Kyungpook University, Daegu 702-701, Korea


In this paper we study that the q-Bernoulli polynomial, which were constructed by T.Kim, are analytic continued to βs(z). A new formula for the q-Riemann Zeta function ζ,(s) due to T.Kim (see [1,2,8]) in terms of nested series of ζ,(n) is derived. The new concept of dynamics of the zeros of analytic continued polynomials is introduced, and an investing phenomenon of ‘scattering’ of the zeros of βs(z) is observed. Following the idea of q-zeta function due to T.Kim, we are going to use “Mathematica” to explore a formula for ζ,(n).


  • q-Bernoulli polynomial
  • q-Riemann Zeta function

AMS Classification

  • 11B68
  • 11S40


  1. T. Kim, S. H. Rim, ‘Generalized Carlitz’s q-Bernoulli Numbers in the
    p-adic number field ’, Adv. Stud. Contemp. Math., 2, 9-19 (2000).
  2. T. Kim, ‘q-Volkenborn integration ’, Russ. J. Math. Phys., 9, 288-299
  3. T. Kim, ‘Non-Arichimedean q-integrals associated with multiple
    Changhee q-Bernoulli polynomials ’, Russ. J. Math. Phys., 10, 91-98
  4. T. Kim, ‘Analytic continuation of multiple q-Zeta functions and their
    values at negative integers ’, to appear in Russ. J. Math. Phys., 11 (2),
  5. T. Kim, ‘A note on multiple zeta functions’, JP J. Algebra, Number
    Theory and Application, 3 (3), 471-476 (2003).
  6. T. Kim, ‘A note on Dirichlet L-series’, Proc. Jangjeon Math. Soc., 6
    (2), 161-166 (2003).
  7. T. Kim, ‘ q-Riemann zeta functions ’, to appear in Int. J. Math. Math.
    Sci., (2004).
  8. T. Kim, ‘On p-adic q-L-function and sums of powers ’, Discrete Math.,
    252, 179-187 (2002).

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Cite this paper

Kim, T., Ryoo, C. S., Jang, L. C., and Rim, S. K. (2005). Exploring the q-Riemann Zeta function and q-Bernoulli polynomials. Notes on Number Theory and Discrete Mathematics, 11(1), 7-19.

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