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Ramanujan’s Formula for ζ(2n + 1)

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Abstract

Ramanujan made many beautiful and elegant discoveries in his short life of 32 years, and one of them that has attracted the attention of several mathematicians over the years is his intriguing formula for ζ(2n + 1). To be sure, Ramanujan’s formula does not possess the elegance of Euler’s formula for ζ(2n), nor does it provide direct arithmetical information. But, one of the goals of this survey is to convince readers that it is indeed a remarkable formula. In particular, we discuss the history of Ramanujan’s formula, its connection to modular forms, as well as the remarkable properties of the associated polynomials. We also indicate analogues, generalizations and opportunities for further research.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana–Champaign, 1409 W. Green St., Urbana, IL, 61801, USA

    Bruce C. Berndt

  2. Department of Mathematics and Statistics, University of South Alabama, 411 University Blvd N, Mobile, AL, 36688, USA

    Armin Straub

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  1. Bruce C. Berndt
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  2. Armin Straub
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Correspondence to Bruce C. Berndt .

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Editors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, Michigan, USA

    Hugh Montgomery

  2. Institut für Mathematik, Universität Zürich, Zürich, Switzerland

    Ashkan Nikeghbali

  3. Institut für Mathematik, Universität Zürich, Zürich, Switzerland

    Michael Th. Rassias

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Berndt, B.C., Straub, A. (2017). Ramanujan’s Formula for ζ(2n + 1). In: Montgomery, H., Nikeghbali, A., Rassias, M. (eds) Exploring the Riemann Zeta Function. Springer, Cham. https://doi.org/10.1007/978-3-319-59969-4_2

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