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EVALUATION OF CERTAIN CLASSES OF EISENSTEIN-TYPE SERIES

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  13 March 2009

HIROFUMI TSUMURA
HIROFUMI TSUMURA*
Affiliation:
Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1, Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan (email: tsumura@tmu.ac.jp)
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Abstract

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In this paper, we consider certain classes of Eisenstein-type series involving hyperbolic functions, and prove some formulas for them which can be regarded as relevant analogues of our previous results. We can also regard these formulas as certain generalizations of the famous formulas for the ordinary Eisenstein series given by Hurwitz.

Keywords

Eisenstein seriesLemniscate constantRiemann zeta-function

MSC classification

Secondary: 11M41: Other Dirichlet series and zeta functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 79 , Issue 2 , April 2009 , pp. 239 - 247
Copyright
Copyright © Australian Mathematical Society 2009

References

[1] Ayoub, R., ‘Euler and the zeta function’, Amer. Math. Monthly 81 (1974), 1067–1086.CrossRefGoogle Scholar
[2] Berndt, B. C., ‘Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan’, J. Reine Angew Math. 303/304 (1978), 332–365.Google Scholar
[3] Berndt, B. C., Ramanujan’s Notebooks, part II (Springer, New York, 1989).CrossRefGoogle Scholar
[4] Berndt, B. C., Ramanujan’s Notebooks, part V (Springer, New York, 1998).CrossRefGoogle Scholar
[5] Cauchy, A., Oeuvres Complètes D’Augustin Cauchy, Série II, Tome VII (Gauthier-Villars, Paris, 1889).Google Scholar
[6] Hurwitz, A., ‘Mathematische Werke. Bd. I: Funktionentheorie’, in: Herausgegeben von der Abteilung für Mathematik und Physik der Eidgenössischen Technischen Hochschule in Zürich (Birkhäuser, Basel, 1962).Google Scholar
[7] Lemmermeyer, F., Reciprocity Laws: From Euler to Eisenstein (Springer, Berlin, 2000).CrossRefGoogle Scholar
[8] Mellin, H., ‘Eine Formel für den Logarithmus transcendenter Funktionen von endlichen Geschlecht’, Acta Soc. Sci. Fennicae 29 (1902), 165–183.Google Scholar
[9] Tsumura, H., ‘On functional relations between the Mordell–Tornheim double zeta functions and the Riemann zeta function’, Math. Proc. Cambridge Philos. Soc. 142 (2007), 395–405.CrossRefGoogle Scholar
[10] Tsumura, H., ‘On certain analogues of Eisenstein series and their evaluation formulas of Hurwitz type’, Bull. London Math. Soc. 40 (2008), 85–93.CrossRefGoogle Scholar