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Ramanujan’s Formula and Modular Forms

  • Chapter
Number Theoretic Methods

Part of the book series: Developments in Mathematics ((DEVM,volume 8))

Abstract

In the theory of zeta-functions, which are defined wherever there are defined norms or substitutes thereof, the ingredients — modular relations, functional equations, incomplete gamma series, and the like — are placed like nodes on the woofs. Some of them are woven by warps as Hecke theory or Lavrik’s theory. The former connects the modular relation to the functional equation, thus making it possible to go to and from between the more orderly world of automorphic forms and the less orderly one of zeta-functions while the latter relates functional equations and incomplete gamma series in the same vein, the idea originating from Riemann. We have found a warp stitching all of these nodes-ingredients, enabling us to warp from one node to another as well as providing us with a guiding principle to locate the exact position and direction of research, a guiding thread to give a clear picture of the whole scene through opaque mist of complexity. We shall illustrate the principle by examples of various zeta-functions satisfying Hecke’s functional equation, i.e. the one with a single gamma factor, in which category many of the important zeta-functions are contained, notably, the Riemann zeta-, Dirichlet L-, Epstein zeta-, the automorphic zeta-functions, etc. In particular, we shall be concerned with the automorphic zeta-functions, the zeta functions arising from automorphic forms, evaluating their special values and obtaining incomplete gamma series.

The authors are supported by Grant-in-Aid for Scientific Research No. 14540051, 14540021 and 14005245 respectively.

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  1. Graduate School of Advanced Technology, University of Kinki, Iizuka, Fukuoka, 820-8555, Japan

    Shigeru Kanemitsu

  2. Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan

    Yoshio Tanigawa & Masami Yoshimoto

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  1. Graduate School of Advanced Technology, University of Kinki, Iizuka, Japan

    Shigeru Kanemitsu

  2. Institute of Mathematics, Academia Sinica, Beijing, China

    Chaohua Jia

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Dedicated to Professor Jonas Kubilius on his eightieth birthday

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© 2002 Springer Science+Business Media Dordrecht

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Kanemitsu, S., Tanigawa, Y., Yoshimoto, M. (2002). Ramanujan’s Formula and Modular Forms. In: Kanemitsu, S., Jia, C. (eds) Number Theoretic Methods. Developments in Mathematics, vol 8. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3675-5_10

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  • DOI: https://doi.org/10.1007/978-1-4757-3675-5_10

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-5239-4

  • Online ISBN: 978-1-4757-3675-5

  • eBook Packages: Springer Book Archive

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