An Alternative Approach to Computing β(2k+1)

2023/09/22 by Tanabe, Naomi, Wattanawanichkul, Nawapan
#11M06 (Primary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2309.13134

Abstract

This paper presents a new approach to evaluating the special values of the Dirichlet beta function, β(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 β(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 ζ(2k).

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