1 The evaluation of character Euler double sums J.M. Borwein*, I.J. Zucker† and J. Boersma‡ *Faculty of Computing Science, Dalhousie University, Halifax, NS, B3H 3J5 Canada. Research supported by NSERC and by the Canada Research Chairs programme. † Wheatstone Physics Laboratory, King’s College, Strand, London WC2R 2LS, UK. ‡Dept of Mathematics and Computing Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands. Abstract Euler considered sums of the form ∞  m=1 1 m s m−1  n=1 1 n t . Here natural generalizations of these sums namely [p,q] := [p,q](s,t)= ∞  m=1 χ p (m) m s m−1  n=1 χ q (n) n t , are investigated, where χ p and χ q are characters, and s and t are positive integers. The cases when p and q are either 1, 2a, 2b or −4 are examined in detail, and closed-form expressions are found for t = 1 and general s in terms of the Riemann zeta function and the Catalan zeta function — the Dirichlet series L −4 (s)=1 −s − 3 −s +5 −s − 7 −s + .... Some results for arbitrary p and q are obtained as well.