Certain classes of series involving the zeta function. (English) Zbl 0932.11054

The authors apply the theory of the double Gamma function, which was recently revived in the study of the determinants of Laplacians, to evaluate some families of series involving the Riemann zeta-function. Introducing a (presumably new) mathematical constant in the theory of the double Gamma function, they also systematically evaluate a definite integral of the double Gamma function and various classes of series associated with Zeta functions. Some of these definite integrals are expressed in terms of quotients of double Gamma functions.


11M06 \(\zeta (s)\) and \(L(s, \chi)\)
33B99 Elementary classical functions
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[1] Ahlfors, L. V., Complex Analysis (1979), McGraw-Hill: McGraw-Hill New York
[2] Alexeiewsky, W., Über eine Classe von Funktionen, die der Gammafunktion analog sind, Leipzig Weidmannsche Buchhandluns, 46, 268-275 (1894)
[3] Barnes, E. W., The theory of the \(G\), Quart. J. Math., 31, 264-314 (1899) · JFM 30.0389.02
[4] Barnes, E. W., Genesis of the double gamma function, Proc. London Math. Soc., 31, 358-381 (1900) · JFM 30.0389.03
[5] Barnes, E. W., The theory of the double gamma function, Philos. Trans. Roy. Soc. London Ser. A, 196, 265-388 (1901) · JFM 32.0442.02
[6] Cassou-Noguès, P., Analogues \(p\), Journées Arithmétiques de Luminy. Journées Arithmétiques de Luminy, Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978 (1979), Soc. Math. France: Soc. Math. France Paris, p. 43-55 · Zbl 0425.12018
[7] Choi, J., Determinant of Laplacian on \(S^3\), Math. Japon., 40, 155-166 (1994) · Zbl 0806.58053
[8] Choi, J., A duplication formula for the double gamma function \(Γ_2\), Bull. Korean Math. Soc., 33, 289-294 (1996) · Zbl 0859.33002
[9] Choi, J.; Srivastava, H. M., Sums associated with the zeta function, J. Math. Anal. Appl., 206, 103-120 (1997) · Zbl 0869.11067
[10] Choi, J.; Srivastava, H. M.; Quine, J. R., Some series involving the zeta function, Bull. Austral. Math. Soc., 51, 383-393 (1995) · Zbl 0830.11030
[11] De Lillo, N. J., Advanced Calculus with Applications (1982), Macmillan: Macmillan New York · Zbl 0484.26004
[12] Edwards, J., A Trestise on the Integral Calculus with Applications, Examples and Problems (1954), Chelsea: Chelsea New York
[13] Gradshteyn, I. S.; Ryzhik, I. M., Tables of Integrals, Series, and Products (1980), Academic: Academic New York · Zbl 0521.33001
[14] Hölder, V. O., Über eine Transcendente Funktion (1886), Dieterichsche Verlags-Buchhandlung: Dieterichsche Verlags-Buchhandlung Göttingen, p. 514-522 · JFM 18.0376.01
[15] Ivić, A., The Riemann Zeta-Function (1985), Wiley: Wiley New York · Zbl 0583.10021
[16] Kinkelin, V. H., Über eine mit der Gamma Funktion verwandte Transcendente und deren Anwendung auf die Integralrechnung, J. Reine Angew. Math., 57, 122-158 (1860)
[17] Magnus, W.; Oberhettinger, F.; Soni, R. P., Formulas and Theorems for the Special Functions of Mathematical Physics (1966), Springer-Verlag: Springer-Verlag New York · Zbl 0143.08502
[18] Matsumoto, K., Asymptotic series for double zeta, double gamma, and Hecke \(L\), Math. Proc. Cambridge Philos. Soc., 123, 385-405 (1998) · Zbl 0903.11021
[19] Osgood, B.; Phillips, R.; Sarnak, P., Extremals of determinants of Laplacians, J. Funct. Anal., 80, 148-211 (1988) · Zbl 0653.53022
[20] Quine, J. R.; Choi, J., Zeta regularized products and functional determinants on spheres, Rocky Mountain J. Math., 26, 719-729 (1996) · Zbl 0864.47024
[21] Shintani, T., A proof of the classical Kronecker limit formula, Tokyo J. Math., 3, 191-199 (1980) · Zbl 0462.10014
[22] Spiegel, M. R., Mathematical Handbook (1968), McGraw-Hill: McGraw-Hill New York
[23] Srivastava, H. M., A unified presentation of certain classes of series of the Riemann zeta function, Riv. Mat. Univ. Parma (4), 14, 1-23 (1988) · Zbl 0659.10047
[24] Titchmarsh, E. C., The Theory of the Riemann Zeta-Function (1951), Oxford University (Clarendon): Oxford University (Clarendon) Oxford · Zbl 0042.07901
[25] Vardi, I., Determinants of Laplacians and multiple gamma functions, SIAM J. Math. Anal., 19, 493-507 (1988) · Zbl 0641.33003
[26] Voros, A., Special functions, spectral functions and the Selberg zeta function, Comm. Math. Phys., 110, 439-465 (1987) · Zbl 0631.10025
[27] Whittaker, E. T.; Watson, G. N., A Course of Modern Analysis (1963), Cambridge University: Cambridge University Cambridge · Zbl 0108.26903
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