The evaluation of character Euler double sums

Abstract

Euler considered sums of the form

$$\sum_{m=1}^{\infty}\frac{1}{m^{s}}\sum_{n=1}^{m-1}\frac{1}{n^{t}}.$$

Here natural generalizations of these sums namely

$$[p,q]:=[p,q](s,t)=\sum_{m=1}^{\infty}\frac{\chi_{p}(m)}{m^{s}}\sum_{n=1}^{m-1}\frac{\chi_{q}(n)}{n^{t}},$$

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.