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A new method in the study of Euler sums. (English) Zbl 1155.40002

Summary: A new method in the study of Euler sums is developed. A host of Euler sums, typically of the form \(\sum_{n=1}^{\infty}\frac{f(n)}{n^{s}}\sum_{m=1}^{n}\frac{g(m)}{m^{t}}\), are expressed in closed form. Also obtained as a by-product are some striking recursive identities involving several Dirichlet series including the well-known Riemann zeta function.

MSC:

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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References:

[1] Apostol, T.M., Vu, T.H.: Dirichlet Series related to the Riemann Zeta function. J. Number Theory 19, 85–120 (1984) · Zbl 0539.10032 · doi:10.1016/0022-314X(84)90094-5
[2] Bailey, D.H., Borwein, J.M., Girgensohn, R.: Experimental evaluation of Euler sums. Exp. Math. 3, 17–30 (1994) · Zbl 0810.11076
[3] Basu, A., Apostol, Tom M.: A new method for investigating Euler sums. Ramanujan J. 4, 397–419 (2000) · Zbl 0971.40001 · doi:10.1023/A:1009868016412
[4] Crandall, R.E., Buhler, J.P.: On the evaluation Euler sums. Exp. Math. 3, 275–285 (1994) · Zbl 0833.11045
[5] Ramanujan, S.: Note Books, vol. 2 (1957) · Zbl 0077.06401
[6] Williams, G.T.: A method of evaluating {\(\zeta\)}(2n). Am. Math. Mon. 60, 19–25 (1953) · Zbl 0050.06803 · doi:10.2307/2306473
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