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Explicit evaluation of Euler sums. (English) Zbl 0819.40003
In response to a letter from Goldbach, Euler considered sums of the form \[ \sigma_ h (s,t):= \sum_{n=1}^ \infty \Biggl(1+ {1\over {2^ s}}+ \dots+ {1\over {(n-1)^ s}} \Biggr) n^{-t}, \] where \(s\) and \(t\) are positive integers.
As Euler discovered by a process of extrapolation (from \(s+t\leq 13\)), \(\sigma_ h (s,t)\) can be evaluated in terms of Riemann \(\zeta\)- functions when \(s+t\) is odd. We provide a rigorous proof of Euler’s discovery and then give analogous evaluations with proofs for corresponding alternating sums. Relatedly we give a formula for the series \[ \sum_{n=1}^ \infty \bigl(1+ {\textstyle {{1\over 2}+\dots+ {1\over n}}}\bigr)^ 2 (n+1)^{-m}. \] This evaluation involves \(\zeta\)- functions and \(\sigma_ h (2,m)\).

40A25 Approximation to limiting values (summation of series, etc.)
40B05 Multiple sequences and series
11M99 Zeta and \(L\)-functions: analytic theory
33E99 Other special functions
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[1] DOI: 10.1016/0377-0427(91)90112-W · Zbl 0782.33001
[2] Berndt, Ramanujan’s Notebooks, Part I (1985) · Zbl 0555.10001
[3] Bailey, Experiment. Math. 3 pp 17– (1994) · Zbl 0810.11076
[4] Stromberg, An Introduction to Classical Real Analysis (1981) · Zbl 0454.26001
[5] Euler, Opera Omnia, Ser 1 XV pp 217– (1917)
[6] DOI: 10.1016/0022-314X(87)90012-6 · Zbl 0606.10032
[7] Nielsen, Die Gammafunktion (1965)
[8] Lewin, Polylogarithms and Associated Functions (1981)
[9] Hoffman, Pacific J. Math. 152 pp 275– (1992) · Zbl 0763.11037
[10] Rao, Pacific J. Math. 113 pp 471– (1984) · Zbl 0549.10031
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