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The evaluation of character Euler double sums. (English) Zbl 1241.11108

Summary: 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 \(\chi_p\) and \(\chi_q\) 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 \(L_{-4}(s)=1^{-s} -3^{-s} +5^{-s} -7^{-s} +\cdots\). Some results for arbitrary \(p\) and \(q\) are obtained as well.

MSC:

11M41 Other Dirichlet series and zeta functions
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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