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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.

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