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Multiple harmonic series. (English) Zbl 0763.11037
The author studies series of the form $$\sum n_ 1^{-i_ 1} n_ 2^{-i_ 2}\cdots n_ k^{-i_ k}$$, where the summation is extended over integers $$n_ 1>n_ 2>\cdots n_ k\geq 1$$ and the exponents satisfy $$i_ j\geq 1$$ and $$i_ 1>1$$. The special case $$k=1$$ is the Riemann zeta function; the case $$k=2$$ has been studied by many authors, beginning with Euler. It is shown that these series are related to products of values of the Riemann zeta function. Conjectures are made for identities that may hold in the case of positive integer exponents $$i_ j$$.

MSC:
 11M41 Other Dirichlet series and zeta functions
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