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Unified Theory of Zeta-Functions Allied to Epstein Zeta-Functions and Associated with Maass Forms

Author

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  • Nianliang Wang

    (College of Applied Mathematics and Computer Science, Shangluo University, Shangluo 726000, China
    Zum Professor Dr. Christopher Deninger für seinen 65sten Geburztag gewidmet mit freunlichen Grüssen und gründlichen Respekt.
    These authors contributed equally to this work.)

  • Takako Kuzumaki

    (Faculty of Engineering, Gifu University, Gifu 501-1193, Japan
    Zum Professor Dr. Christopher Deninger für seinen 65sten Geburztag gewidmet mit freunlichen Grüssen und gründlichen Respekt.
    These authors contributed equally to this work.)

  • Shigeru Kanemitsu

    (KSCSTE-Kerala School of Mathematics, Kunnamangalam, Kozhikode 673571, Kerala, India
    Zum Professor Dr. Christopher Deninger für seinen 65sten Geburztag gewidmet mit freunlichen Grüssen und gründlichen Respekt.
    These authors contributed equally to this work.)

Abstract

In this paper, we shall establish a hierarchy of functional equations (as a G -function hierarchy) by unifying zeta-functions that satisfy the Hecke functional equation and those corresponding to Maass forms in the framework of the ramified functional equation with (essentially) two gamma factors through the Fourier–Whittaker expansion. This unifies the theory of Epstein zeta-functions and zeta-functions associated to Maass forms and in a sense gives a method of construction of Maass forms. In the long term, this is a remote consequence of generalizing to an arithmetic progression through perturbed Dirichlet series.

Suggested Citation

  • Nianliang Wang & Takako Kuzumaki & Shigeru Kanemitsu, 2023. "Unified Theory of Zeta-Functions Allied to Epstein Zeta-Functions and Associated with Maass Forms," Mathematics, MDPI, vol. 11(4), pages 1-24, February.
  • Handle: RePEc:gam:jmathe:v:11:y:2023:i:4:p:917-:d:1065297
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