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A decomposition of Riemann’s zeta-function. (English) Zbl 0907.11024
Motohashi, Y. (ed.), Analytic number theory. Proceedings of the 39th Taniguchi international symposium on mathematics, Kyoto, Japan, May 13--17, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 247, 95-101 (1997).
Let $$\zeta(p_{1},\dots,p_g) := \sum_{a_{1}>\dots>a_{g}\ge 1} \frac{1}{a_{1}^{p_{1}}} \cdots \frac{1}{a_{g}^{p_{g}}},$$ where $a_i,p_i$ are positive integers with $p_1\ge 2$. It was conjectured independently by {\it M. Hoffman} [Pac. J. Math. 152, 275-290 (1992; Zbl 0763.11037)] and by {\it C. Markett} [J. Number Theory 48, 113-132 (1994; Zbl 0810.11047)] that $$\sum_{p_{1}+\dots+p_{g}=N} \zeta(p_{1},\dots,p_{g})= \zeta(N)$$ for positive integers $g,N$ with $N\ge g+1$. This conjecture is proved in the present paper by generating functions arguments. The author then uses this result to derive formulas for certain double and triple sums (proved earlier by different methods). For the entire collection see [Zbl 0874.00035].

11M06$\zeta (s)$ and $L(s, \chi)$
05A19Combinatorial identities, bijective combinatorics
11M41Other Dirichlet series and zeta functions
05A15Exact enumeration problems, generating functions