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\(q\)-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.)

Citations:

Zbl 0501.12020
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References:

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