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Zbl 0907.11024
Granville, Andrew
A decomposition of Riemann's zeta-function.
(English)
[A] 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). ISBN 0-521-62512-2/pbk

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).
[R.Girgensohn (Neuherberg)]
MSC 2000:
*11M06 Riemannian zeta-function and Dirichlet L-function
05A19 Combinatorial identities
11M41 Other Dirichlet series and zeta functions
05A15 Combinatorial enumeration problems

Keywords: Riemann zeta function; multiple harmonic series; Euler-Zagier sums; generating functions; combinatorial identities

Citations: Zbl 0763.11037; Zbl 0810.11047

Cited in: Zbl 1229.11117 Zbl 1156.11038

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