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## Euler Sums

Euler sums (also called multiple zeta values'' or multiple harmonic sums'') are a useful generalization of the classical zeta function: For positive integers we define

 (7)

and we note that i1>1 is both necessary and sufficient condition for convergence. Thence we get the mapping . The integer k is called the depth of the sum, and is called the weight of the sum. The definition (7) can be extended to alternating sums. Euler sums recently found very interesting interpretations in knot theory [19,20,21,22], and high-energy particle physics [17] (quantum field theory).

Euler sums satisfy many striking identities, of which is the simplest one. (Let us remark that Euler himself suggested and partially proved theorems about reducibility of depth 2 zetas to depth 1 zetas.) Examples of such depth 2 identities, derived algebraically, can be found in [34], Appendix B. Recent work on integral representations of Euler sums has led us to a fast algorithm for their high precision evaluations [13], thus allowing us to thoroughly examine them using integer relation algorithms.

The example that we want to single out for presentation is motivated by the identity conjectured by Zagier [37] and proved first by Broadhurst [13], after extensive empirical work in [12]:

where denotes the string s repeated n times. Let us mention in this context that, in all known non-decomposable'' identities involving Euler sums, all -terms have the same weight. This fact of course is of great importance for guided integer relation searches--as it dramatically reduces the size of the search space.

Broadhurst and Lisonek recently used Bailey's fast implementation of the PSLQ algorithm (Section 2.2) to search for possible generalizations of Zagier's identity. Soon it appeared that the values

participate in many such identities. We then started to form the PSLQ input vectors from all Z values of fixed weight (2K, say) along with the value ; and were rewarded by detecting many identities, from which we were able to infer general patterns, and conjecture (among many other things) the following generalization of Zagier's identity [14]:

 (8)

where S is a string of 2n+1 non-negative integers whose sum is M, and Ci S denotes the cyclic shift of S by i positions. Zagier's identity is a special case of (8) with all entries of S being zeros.

The symmetric form of (8) highlighted that Zagier-type identities have a lot of combinatorial content, and in the cases M=0,1 we were able to reduce (8) to evaluation of certain combinatorial sums, yielding in this way combinatorial proofs whose only analytic component is the known representation of Euler sums as iterated integrals [37]. In the cases we do not have such proofs yet, but we have strong numerical evidence supporting (8) in these cases also.

The conjecture (8)--which has been proved in [14] to the extent mentioned above--would never have been identified without extensive use of an integer relation algorithm. Not only did we infer new identities as results of sophisticated PSLQ runs, but we even received a different perspective on the entire situation and new avenues and proof styles opened up for us.

Next: The Prouhet-Tarry-Escott Problem Up: Applications Previous: Cubic Singular Values
Agnes Szanto
2000-05-10