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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