An Alternative Approach to Computing $\beta(2k+1)$

Naomi Tanabe; Nawapan Wattanawanichkul

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arXiv · 2309.13134

An Alternative Approach to Computing $\beta(2k+1)$

Naomi Tanabe·Nawapan Wattanawanichkul

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Abstract

This paper presents a new approach to evaluating the special values of the Dirichlet beta function, $\beta(2k+1)$, where $k$ is any nonnegative integer. Our approach relies on some properties of the Euler numbers and polynomials, and uses basic calculus and telescoping series. By a similar procedure, we also yield an integral representation of $\beta(2k)$. The idea of our proof adapts from a previous study by Ciaurri et al., where the authors introduced a new proof of Euler's formula for $\zeta(2k)$.

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Naomi Tanabe, Nawapan Wattanawanichkul. 2023-09-22. An Alternative Approach to Computing $\beta(2k+1)$. https://arxiv.org/abs/2309.13134

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