Journal of the
Korean Mathematical Society JKMS

In this presentation dedicated to the tricentennial birth anniversary of the great eighteenth-century Swiss mathematician, Leonhard Euler (1707-1783), we begin by remarking about the so-called $Basler$ $problem$ of evaluating the Zeta function $\zeta \left( s\right)$ [in the much later notation of Georg Friedrich Bernhard Riemann (1826-1866)] when $s=2$, which was then of vital importance to Euler and to many other contemporary mathematicians including especially the Bernoulli brothers [Jakob Bernoulli (1654-1705) and Johann Bernoulli (1667-1748)], and for which a fascinatingly large number of seemingly independent solutions have appeared in the mathematical literature ever since Euler first solved this problem in the year 1736. We then investigate various recent developments on the evaluations and representations of $\zeta \left( s\right)$ when $s \in \mathbb{N}\,\backslash \left\{ 1\right\}$, $\mathbb{N}$ being the set of natural numbers. We emphasize upon several interesting classes of rapidly convergent series representations for $\zeta \left( 2n+1\right) $ $\left( n\in \mathbb{N}\right)$ which have been developed in recent years. In two of many computationally useful special cases considered here, it is observed that $\zeta \left( 3\right)$ can be represented by means of series which converge much more rapidly than that in Euler's celebrated formula as well as the series used recently by Roger Ap\'{e}ry (1916-1994) in his proof of the irrationality of $\zeta \left( 3\right) $. Symbolic and numerical computations using $Mathematica$ (Version 4.0) for Linux show, among other things, that only 50 terms of one of these series are capable of producing an accuracy of seven decimal places.

Keywords: analytic number theory, Riemann zeta function, Hurwitz (or generalized) zeta function, series representations, harmonic numbers, Bernoulli numbers and polynomials, generating functions, Euler numbers and polynomials, inductive argument, symbolic and numer

MSC numbers: Primary 11M06, 11M35, 33B15; Secondary 11B68, 33E20, 33E30