Borwein, J. M.; Zucker, I. J.; Boersma, J. The evaluation of character Euler double sums. (English) Zbl 1241.11108 Ramanujan J. 15, No. 3, 377-405 (2008). 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. Cited in 1 ReviewCited in 42 Documents MSC: 11M41 Other Dirichlet series and zeta functions 11M06 \(\zeta (s)\) and \(L(s, \chi)\) Keywords:Euler sums; Dirichlet characters; Riemann zeta function PDFBibTeX XMLCite \textit{J. M. Borwein} et al., Ramanujan J. 15, No. 3, 377--405 (2008; Zbl 1241.11108) Full Text: DOI Online Encyclopedia of Integer Sequences: Decimal expansion of L_2 = -Integral_{x=0..Pi/2} log(2*sin(x/2))^2 dx, a constant appearing in the evaluation of Euler double sums not expressible in terms of well-known constants. Decimal expansion of J_4 = Integral_{0..Pi/2} x^4/sin(x) dx. Decimal expansion of J_5 = Integral_{0..Pi/2} x^5/sin(x) dx. References: [1] Basu, A., Apostol, T.M.: A new method for investigating Euler sums. Ramanujan J. 4, 397–419 (2000) · Zbl 0971.40001 · doi:10.1023/A:1009868016412 [2] Borwein, J., Bailey, D.: Mathematics by Experiment: Plausible Reasoning in the 21st Century. AK Peters, Natick (2003) · Zbl 1163.00002 [3] Borwein, J., Bailey, D., Girgensohn, R.: Experimentation in Mathematics: Computational Paths to Discovery. AK Peters, Natick (2004) · Zbl 1083.00002 [4] Euler, L.: Meditationes circa singulare serierum genus. Novi Commun. Acad. Sci. Petropol. 20, 140–186 (1775) [5] Jordan, P.F.: Infinite sums of psi functions. Bull. Am. Math. Soc. 79, 681–683 (1973) · Zbl 0266.33011 · doi:10.1090/S0002-9904-1973-13259-8 [6] Lewin, L.: Polylogarithms and Associated Functions. North-Holland, New York (1981) · Zbl 0465.33001 [7] Nielsen, N.: Die Gammafunktion. Chelsea, New York (1965) [8] Sitaramachandrarao, R.: A formula of S. Ramanujan. J. Number Theory 25, 1–19 (1987) · Zbl 0606.10032 · doi:10.1016/0022-314X(87)90012-6 [9] Terhune, D.: Evaluations of double L-values. J. Number Theory 105, 275–301 (2004) · Zbl 1041.11061 · doi:10.1016/j.jnt.2003.11.007 [10] Zucker, I.J., Robertson, M.M.: Some properties of Dirichlet L-series. J. Phys. A: Math. Gen. 9, 1207–1214 (1976) · Zbl 0338.10037 · doi:10.1088/0305-4470/9/8/006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.