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The 6-order sums of Riemann zeta function. (Chinese. English summary) Zbl 1121.11309
Summary: We study in this paper certain series involving \(\zeta_n(k)\), which are the partial sums of Riemann-zeta function \(\zeta (k)\), by the probabilistic and combinatorial methods, several important sums are evaluated. Specially the known result of Euler \(5\zeta (4)=2\zeta(2)\) can be derived directly from the three sums of 4-order, and therefore the eleven sums of 6-order evaluated in this paper imply that it is possible to obtain the value of \(\zeta (3)\) from searching for the nontrivial relation among certain series.

MSC:
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
05A19 Combinatorial identities, bijective combinatorics
65B10 Numerical summation of series
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