A simple proof of Linas’s theorem on Riemann zeta function

Jun Ikeda, Junsei Kochiya and Takato Ui
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 4, Pages 90–94
DOI: 10.7546/nntdm.2021.27.4.90-94
Full paper (PDF, 142 Kb)

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Authors and affiliations

Jun Ikeda
Kaijo School, Shinjuku, Tokyo, Japan

Junsei Kochiya
Kaijo School, Shinjuku, Tokyo, Japan

Takato Ui
Kaijo School, Shinjuku, Tokyo, Japan

Abstract

Linas Vepštas gives rapidly converging infinite representatives for values of Riemann zeta function at \left(4m-1 \right), where m is a natural number. In this paper, we give a new simple proof. Also, we obtain two equation of values of Bernoulli numbers’ generating function by applying a corollary given in this paper.

Keywords

  • Analysis
  • Riemann zeta function
  • Fourier series
  • Hyperbolic function

2020 Mathematics Subject Classification

  • 11M06

References

  1. Glaisher, J. W. L. (1875). On a Class of Identical Relations in the Theory of Elliptic Functions. Philosophical Transactions of the Royal Society of London, 165, 489–518. Available online: http://www.jstor.org/stable/109157
  2. Vepštas, L. (2012). On Plouffe’s Ramanujan identities. The Ramanujan Journal, 27(4), 387–408.
  3. Volkovyskii, L. I., Lunts, G. L., & Aramanovich, I. G. (1965). A Collection of Problems on Complex Analysis, Dover Publications, Inc., New York.

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Cite this paper

Ikeda, J., Kochiya, J., & Ui, T. (2021). A simple proof of Linas’s theorem on Riemann zeta function. Notes on Number Theory and Discrete Mathematics, 27(4), 90-94, DOI: 10.7546/nntdm.2021.27.4.90-94.

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