Experimental Evaluation of Euler Sums

Abstract

CONTENTS 1. Introduction 2. Numerical Techniques 3. Experimental Setup and Optimizations 4. Integer Relation Detection Algorithms 5. Applications of the PSLQ Algorithm 6. Experimental Results 7. Conjectures Acknowledgments References Borwein was supported by NSERC and the Shrum Endowment at Simon Fraser University. Girgensohn was supported by a DFG fellowship. Euler expressed certain sums of the form 1 X k=1 1 + 1 2 m + + 1 k m (k + 1) ;n , where m and n are positive integers, in terms of the Riemann zeta function. In [Borwein et al. 1993], Euler's results were extended to a significantly larger class of sums of this type, including sums with alternating signs. This research was facilitated by numerical computations using an algorithm that can determine, with high confidence, whether or not a particular numerical value can be expressed as a rational linear combination of several given constants. The present paper presents the numerical techniques used in these computations and lists many of the experimental results that have been obtained.

Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics

“…[David Bailey] then computed this constant to 100 decimal digits, and the above equality was still affirmed." Many formulas similar to (43) have subsequently been established by rigorous proof [50].…”

Explicit evaluation of Euler sums

“…(x,3>):=f(l-u)'-V~1<fM = r^r^ for x>0,. In this paper, we will consider s h (2, m), leaving discussion of the more general s b (n, m) for [1]. The purpose of this paper is to give a complete proof of the formulas This last evaluation is due to Euler (cf.…”

Fast evaluation of multiple zeta sums

“…The multiple zeta sums: [3], Borwein et al [4], Borwein et al [7], Broadhurst et al [9], Crandall and Buhler [11], Markett [13] and Zagier [14]) (note that some previous treatments have reversed ordering of the indices n i ). The study of such sums is not only important to general zeta function theory, but also touches upon such domains as knot theory and particle physics methodology.…”

“…(where n denotes the the number of (3, 1) pairs) remain elusive. * Even the twodimensional sums ζ(6, 2) and ζ (3,5) are unknown, in the sense that neither has been cast in a finite form involving "one-dimensional" sums such as Riemann zeta values. There is a growing literature on which sums are evaluable, which can be reduced to lower-dimensional forms, and which appear dimensionally irreducible and therefore "fundamental."…”

“…The requirement of high precision arises for various reasons, such as the desire to avoid "accidental" relations and to guard against mishap when some integral coefficients of exact relations are large. The original approach of Bailey et al [3] involved an Euler-Maclaurin scheme for estimating double sums. This approach has been effectively generalized by Broadhurst, as described in [7], to higher dimensions.…”