Rapidly Converging Series for ζ(2n+1)‎ from Fourier Series

Abstract EN

Ever since Euler first evaluated ζ(2) and ζ(2m), numerous interesting solutions of the problem of evaluating the ζ(2m) (m∈ℕ) have appeared in the mathematical literature.

Until now no simple formula analogous to the evaluation of ζ(2m) (m∈ℕ) is known for ζ(2m+1) (m∈ℕ) or even for any special case such as ζ(3).

Instead, various rapidly converging series for ζ(2m+1) have been developed by many authors.

Here, using Fourier series, we aim mainly at presenting a recurrence formula for rapidly converging series for ζ(2m+1).

In addition, using Fourier series and recalling some indefinite integral formulas, we also give recurrence formulas for evaluations of β(2m+1) and ζ(2m) (m∈ℕ), which have been treated in earlier works.