## $$q$$-analogue of Riemann’s $$\zeta$$-function and $$q$$-Euler numbers.(English)Zbl 0675.12010

N. Koblitz [J. Number Theory 14, 332–339 (1982; Zbl 0501.12020)] has given a $$q$$-analogue of the $$p$$-adic $$L$$-function, and asked whether there is a corresponding complex analytic $$q$$-$$L$$-function. The author answers this by giving
$L_q(s,\chi) = \frac{2-s}{s-1}(q-1)\sum^{\infty}_{n=1}\frac{q^n\chi(n)}{[n]^{s- 1}}+\sum_{n=1}\frac{q^n\chi (n)}{[n]^s},$
where $$[n]=(1-q^n)/(1-q)$$. This correctly interpolates the $$q$$-Bernoulli numbers at negative integers. As an application the author proves Kummer congruences for $$q$$-Euler numbers.

### MSC:

 11S40 Zeta functions and $$L$$-functions 11B68 Bernoulli and Euler numbers and polynomials 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.)

Zbl 0501.12020
Full Text:

### References:

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