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Experimental evaluation of Euler sums. (English) Zbl 0810.11076
The authors consider eight classes of infinite series they call Euler sums. Two representative examples are \[ \sum_{k=1}^ \infty \bigl( 1+ {\textstyle {1\over 2}}+ \cdots+ {\textstyle {1\over k}} \bigr)^ m (k+1)^{-n} \quad \text{ and } \quad \sum_{k=1}^ \infty \bigl(1+ {\textstyle {1\over {2^ m}}} +\cdots+ {\textstyle {1\over {k^ m}}} \bigr) (k+1)^{-n}, \] where \(m\), \(n\) are positive integers with \(n\geq 2\); the other six are variants with alternating signs. A method is described for high-precision numerical evaluation of all these sums, and some of them are evaluated explicitly in closed form in terms of the Riemann zeta function. Many of the closed-form identities were detected experimentally. It is an open question whether such closed-form identities exist for all the Euler sums considered in this paper.

MSC:
11Y60 Evaluation of number-theoretic constants
40A25 Approximation to limiting values (summation of series, etc.)
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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