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Rapidly Converging Series for from Fourier Series

DOI: 10.1155/2014/457620

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Ever since Euler first evaluated and , numerous interesting solutions of the problem of evaluating the have appeared in the mathematical literature. Until now no simple formula analogous to the evaluation of ?? is known for or even for any special case such as . Instead, various rapidly converging series for have been developed by many authors. Here, using Fourier series, we aim mainly at presenting a recurrence formula for rapidly converging series for . In addition, using Fourier series and recalling some indefinite integral formulas, we also give recurrence formulas for evaluations of and , which have been treated in earlier works. 1. Introduction and Preliminaries The Riemann zeta function is defined by (see, e.g., [1, p. 164]) The Riemann zeta function in (1) plays a central role in the applications of complex analysis to number theory. The number-theoretic properties of are exhibited by the following result known as Euler's formula, which gives a relationship between the set of primes and the set of positive integers: where the product is taken over all primes. The solution of the so-called Basler problem (cf., e.g., [2], [3, p. xxii], [4, p.66], [5, pp. 197-198], and [6, p. 364]) was first found in 1735 by Euler (1707–1783) [7], although Jakob Bernoulli (1654–1705) and Johann Bernoulli (1667–1748) did their utmost to sum the series in (3). The former of these Bernoulli brothers did not live to see the solution of the problem, and the solution became known to the latter soon after Euler found it (see, for details, Knopp [8, p.238]). Five years later in 1740, Euler (see [9]; see also [10, pp. 137–153]) succeeded in evaluating all of ): where ( ) are the th Bernoulli numbers defined by the following generating function (see, e.g., [1, p. 81]): The following recursion formula can be used for computing Bernoulli numbers. Ever since Euler first evaluated and , numerous interesting solutions of the problem of evaluating the 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 and (see, e.g., [5, 11, 12]), the reader may be referred to the very recent work [13] which contains an extensive literature of as many as more than 70 papers. We may recall here a known recursion formula for (see, e.g., [1, p. 167], [1, Section 4.1], and [14, Theorem I]): which can also be used to evaluate . The eta function or the alternating zeta function is defined by Then it is easy to have the following relation between and

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