Explicit evaluation of Euler sums

Title: Explicit evaluation of Euler sums
Creator: Borwein, David; Borwein, Jonathan M.; Girgensohn, Roland
Relation: Proceedings of the Edinburgh Mathematical Society Vol. 38, Issue 2, p. 277-294
Publisher Link: http://dx.doi.org/10.1017/S0013091500019088
Publisher: Cambridge University Press
Resource Type: journal article
Date: 1995
Description: In response to a letter from Goldbach, Euler considered sums of the form [unable to replicate formula] where s and t are positive integers. As Euler discovered by a process of extrapolation (from s + t ≥ 13), σ_h(s,t) can be evaluated in terms of Riemann ζ-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 [unable to replicate formula]. This evaluation involves ζ-functions and σ_h(2,m).
Subject: Goldbach; Euler sums; Riemann zeta function
Identifier: http://hdl.handle.net/1959.13/1043647
Identifier: uon:14235
Identifier: ISSN:0013-0915
Language: eng
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