# The evaluation of character Euler double sums

Borwein, Jonathan M. and Zucker, I. J. and Boersma, J. (2008) The evaluation of character Euler double sums. Ramanujan Journa, 15 . pp. 377-405.

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Euler considered sums of the form $$\sum_{m=1}^{\infty}\f{1}{m^s}\sum_{n=1}^{m-1}\f{1}{n^t}$$ Here natural generalisations of these sums, namely $$[d_1,d_2]:=[d_1,d_2](s,t)=\sum_{m=1}^{\infty}\f{\rchi_{d_1}(m)}{m^s}\sr\f{\rchi_{d_2}(n)}{n^t},$$ have been investigated, where $\rchi_{d_1}$ and $\rchi_{d_2}$ are characters, and $s$ and $t$ are positive integers. The cases when $d_1$ and $d_2$ were either $1, 2a, 2b$ or $-4$ have been examined in detail, and closed form expressions have been found for $t=1$ and general $s$ in terms of the Riemann zeta function and the Catalan zeta function---the Dirichlet series $L_{-4}(s)=1^{-s}-3^{-s}+5^{-s}-7^{-s}\ldots$. Some results for arbitrary $d_1$ and $d_2$ have also been obtained.
Item Type: Article pubdom FALSE Euler sums, Dirichlet series, zeta functions, character sums 11-xx Number theory > 11Mxx Zeta and $L$-functions: analytic theory40-xx Sequences, series, summability > 40Bxx Multiple sequences and series UNSPECIFIED Users 1 not found. 29 Jan 2004 11 Jan 2015 18:02 https://docserver.carma.newcastle.edu.au/id/eprint/255