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On modification of the \(q\)-\(L\)-series and its applications. (English) Zbl 1025.11030
In an earlier paper [Proc. Japan Acad., Ser. A 75, 23–25 (1999; Zbl 0931.11031)] the author introduced a version of the \(q\)-Riemann \(\zeta\)-function. Now he extends his results introducing and studying the \(q\)-Hurwitz \(\zeta\)-function and \(q\)-\(L\)-series (for another approach see J. Satoh [J. Number Theory 31, 346–362 (1989; Zbl 0675.12010)]). For \(q\to 1\), some relations for the ordinary Dirichlet \(L\)-series are obtained. In particular, a new proof is given for Katsurada’s formula for the values of the Dirichlet \(L\)-series at positive integers [M. Katsurada, Acta Arith. 90, 79–89 (1999; Zbl 0933.11042)].

MSC:
11M41 Other Dirichlet series and zeta functions
11B68 Bernoulli and Euler numbers and polynomials
05A30 \(q\)-calculus and related topics
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