A few remarks on the values of the Bernoulli polynomials at rational arguments and some relations with ζ(2k + 1)

André Pierro de Camargo and Giulliano Cogui de Oliveira Teruya
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 4, Pages 180–186
DOI: 10.7546/nntdm.2021.27.4.180-186
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Authors and affiliations

André Pierro de Camargo
Federal University of the ABC region, Brazil

Giulliano Cogui de Oliveira Teruya
Federal University of the ABC region, Brazil

Abstract

A problem posed by Lehmer in 1938 asks for simple closed formulae for the values of the even Bernoulli polynomials at rational arguments in terms of the Bernoulli numbers. We discuss this problem based on the Fourier expansion of the Bernoulli polynomials. We also give some necessary and sufficient conditions for ζ(2k + 1) be a rational multiple of π^2k+1.

Keywords

• Bernoulli polynomials
• Bernoulli numbers
• Riemann zeta function
• Euler’s formula

2020 Mathematics Subject Classification

• 11B68
• 11M99

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