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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## Details

### 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*(2*k* + 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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## Cite this paper

De Camargo, A. P., & De Oliveira Teruya, G. C. (2021). A few remarks on the values of the Bernoulli polynomials at rational arguments and some relations with *ζ*(2*k* + 1). Notes on Number Theory and Discrete Mathematics, 27(4), 180-186, doi: 10.7546/nntdm.2021.27.4.180-186.