Some series involving the zeta function. (English) Zbl 0830.11030

The knowledge of the double gamma function is applied in order to evaluate some series involving the Riemann zeta function and its generalization, the Hurwitz zeta function. The authors start their considerations from Alexeiwsky’s theorem concerning the integral of the logarithm of the ordinary gamma function (where the double gamma function appears in the result). They extend the theorem conveniently. After performing some analytical manipulations they obtain a series of formulas, some of which they compare with known ones. They finish with some considerations on the \(n\)-ple gamma functions and its uses in the computation of some determinants.


11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M35 Hurwitz and Lerch zeta functions
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[1] Whittaker, A course of modern analysis (1963) · Zbl 0108.26903
[2] Ivić, The Riemann zeta-function (1985)
[3] Gradshteyn, Tables of integrals, series and products (1980) · Zbl 0521.33001
[4] Choi, Math. Japon. 40 pp 155– (1994)
[5] Cassou-Nogués, Analogues p-adiques des fonctions {\(\Gamma\)}-multiples (1978)
[6] DOI: 10.1098/rsta.1901.0006 · JFM 32.0442.02
[7] DOI: 10.1112/plms/s1-31.1.358 · JFM 30.0389.03
[8] Barnes, Quart. J. Math. 31 pp 264– (1899)
[9] Alexejewsky, Leipzig: Weidmanncshe Buchhandluns 46 pp 268– (1894)
[10] Abramowitz, Handbook of mathematical functions (1965)
[11] Vignéras, Soc. Math. France Asté 61 pp 235– (1979)
[12] Titchmarsh, The theory of the Riemann zeta function (1951) · Zbl 0042.07901
[13] Srivastava, Riv. Mat. Univ. Parma 14 pp 1– (1988)
[14] Shintani, Tokyo J. Math. 3 pp 191– (1980)
[15] DOI: 10.2307/2323614
[16] Magnus, Formulas and theorems for the special functions of mathematical physics (1966) · Zbl 0143.08502
[17] Kinkelin, J. Reine Angew. Math. 57 pp 122– (1860) · ERAM 057.1509cj
[18] Jordan, Calculus of finite differences (1965) · Zbl 0154.33901
[19] DOI: 10.1090/S0002-9904-1906-01374-X · JFM 37.0297.03
[20] Hölder, Gottingen Dieterichsche Verlags-Buchhandlung pp 514– (1886)
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