Explicit evaluation of Euler sums

Abstract

In response to a letter from Goldbach, Euler considered sums of the form where s and t are positive integers.As Euler discovered by a process of extrapolation (from s + f g l 3 ) , <r h (s,f) can be evaluated in terms of Riemann (-functions when s + t is odd. We provide a rigorous proof of Euler's discovery and then give analogous evaluations with proofs for corresponding alternating sums. Relatedly we give a formula for the series This evaluation involves {-functions and <r h (2,m).1991 Mathematics subject clasification: Primary 40A25, 4OBO5. Secondary 11M99, 33E99.

Analytic continuation of multiple zeta functions

“…Nevertheless it is our primary goal in this note to define the analytic continuation of the multiple zeta function on all of C n . As a side remark, the special values of multiple zeta functions at positive integers have come to the foreground in recent years ( [2,3,4], etc. ), both in connection with theoretical physics (Feynman diagrams) and the theory of mixed Tate motives.…”

Fast evaluation of multiple zeta sums

“…It could be said that the research situation is quite rich, in the following sense. Whereas Euler's original evaluations, of which ζ(4, 1) = 2ζ(5) − ζ(2)ζ (3) is exemplary, along with more modern evaluations such as: ζ(4, 5, 1) = 2ζ(7, 3) + ζ(5) 2 − 17ζ(3)ζ(7) + ζ(2)ζ(5, 3) + 10ζ(2)ζ(3)ζ(5) − 21 20 ζ(10), ζ(6, 6, 6, 6) = 4π 24 432684797065192546875 , are now proved (Markett [13], Borwein and Girgensohn [8], Borwein et al [7]), suspected evaluations such as Zagier's conjectured: ζ(3, 1, 3, 1, ..., 3, 1) = 2π 4n (4n + 2)! (where n denotes the the number of (3, 1) pairs) remain elusive.…”

“…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.…”

“…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." For example, in Borwein and Girgensohn [8] it is argued that 3-dimensional sums can always be reduced (to lower-dimensional forms) if s 1 + s 2 + s 3 is either even or less than 10. On the other hand, ζ(5, 3, 3) has never been expressed as a combination of, say, two-dimensional ζ sums, "one-dimensional" Riemann sums, and fundamental constants.…”

Exponential generating function of hyperharmonic numbers indexed by arithmetic progressions

“…There is a vast literature dealing with sums involving harmonic numbers, see [4,15,20,30] for example. In contrast, this is not the case for hyperharmonic numbers.…”