## Abstract

Euler considered sums of the form

Here natural generalizations of these sums namely

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 1,2*a*,2*b* 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.

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In Memoriam: Between the submission and acceptance of this report we greatly regret that our esteemed colleague John Boersma passed away. This paper is dedicated to his memory.

This research supported by NSERC and by the Canada Research Chairs programme.

The encouragement and support of Geoff Joyce and Richard Delves at King’s College, London, is much appreciated.

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Borwein, J.M., Zucker, I.J. & Boersma, J. The evaluation of character Euler double sums.
*Ramanujan J* **15**, 377–405 (2008). https://doi.org/10.1007/s11139-007-9083-z

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DOI: https://doi.org/10.1007/s11139-007-9083-z