Rapidly Converging Series forζ(2n+1)from Fourier Series

Abstract

Ever since Euler first evaluatedζ(2)andζ(2m), numerous interesting solutions of the problem of evaluating theζ(2m) (m∈ℕ)have appeared in the mathematical literature. Until now no simple formula analogous to the evaluation ofζ(2m) (m∈ℕ)is known forζ(2m+1) (m∈ℕ)or even for any special case such asζ(3). Instead, various rapidly converging series forζ(2m+1)have been developed by many authors. Here, using Fourier series, we aim mainly at presenting a recurrence formula for rapidly converging series forζ(2m+1). In addition, using Fourier series and recalling some indefinite integral formulas, we also give recurrence formulas for evaluations ofβ(2m+1)andζ(2m) (m∈ℕ), which have been treated in earlier works.

Numerical calculation of the Riemann zeta function at odd-integer arguments: a direct formula method

“…Numerical result show that 20 points are capable of producing an accuracy of seven-decimal place for small arguments. Besides, many rapidly converging series for ζ(2n + 1) have been introduced by Srivastava in a review article [13] and by other authors [12,14].…”

“…where we have used the fact that ζ(s) < , which is superior to other rapid series |R 25 | < 0.9 × 10 −18 as noted in [12,8]. Specially, when N is larger than some typical numbers, the asymptotic behavior of (37) reads (38) lg(|R (s) N |) ∼ −2 lg 2(N + 1) − 3 lg N. The accuracy of the latter two methods rely on the number of zeros(denoted as N 1 ) of the associated polynomial(in this occasion it is Hermite polynomial) and the terms(denoted as N 2 ) of partial sum of the infinite series respectively.…”

“…Meanwhile, the calculation of Riemann zeta function and related series is a hot topic in computational mathematics. The traditional methods are Euler-Maclaurin formula and Riemann-Siegel formula, and algorithms are still being developed in earnest ever since [11,12,13,14]. Typically, a particular numerical method is limited to a special domain.…”

CERTAIN INTEGRAL REPRESENTATIONS OF GENERALIZED STIELTJES CONSTANTS γk(a)

“…which can also be used to evaluate ζ(2n) (n ∈ N \ {1}) by recalling the Basler problem ζ(2) = π 2 /6 (see, e.g., [6] and the references cited therein).…”

Evaluation of Certain Alternating Series

“…Ever since Euler first evaluated ζ(2) and ζ(2n), numerous interesting solutions of the problem of evaluating the ζ(2n) (n ∈ N) have appeared in the mathematical literature. Even though there were certain earlier works which gave a rather long list of papers and books together with some useful comments on the methods of evaluation of ζ(2) and ζ(2n) (see, e.g., [9], [25], [45] and [75]), the reader may be referred to the very recent work [19] which contains an extensive literature of as many as more than 70 papers. Among many different ways to prove (3), several authors have taken advantage of the nice interplay between a double integral and a geometric series (see, e.g., [5,11,39]).…”