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Multiple zeta functions: An example. (English) Zbl 0795.11037
Kurokawa, N. (ed.) et al., Zeta functions in geometry. Tokyo: Kinokuniya Company Ltd.. Adv. Stud. Pure Math. 21, 219-226 (1992).
Given a meromorphic function $$Z_ i(s)$$ of finite order, write $$Z_ i(s)\sim \prod_{\rho\in A_ i} (s-\rho)^{m_ i(\rho)}$$, where $$A_ i$$ is the set of all the zeros and poles of $$Z_ i$$ and $$m_ i(\rho)$$ is equal to the multiplicity of $$\rho$$, $$1\leq i\leq r$$. By definition, the “multiple zeta function” $$Z= Z_ 1\otimes \dots\otimes Z_ r$$ satisfies the relation: $Z(s)\sim \prod_{{\overset \rightharpoonup\rho}\in A} \bigl(s- \sum_{i=1}^ r \rho_ i\bigr)^{m({\overset \rightharpoonup\rho})},$ where $$A= A_ 1\times\dots \times A_ r$$, and $$m({\overset \rightharpoonup\rho})= \sigma ({\overset \rightharpoonup\rho}) \prod_{i=1}^ r m_ i(\rho_ i)$$ with $$\sigma ({\overset \rightharpoonup\rho})=1$$ if $$\text{Im } \rho_ i\geq 0$$ for $$1\leq i\leq r$$, $$\sigma({\overset \rightharpoonup\rho})= (-1)^{r-1}$$ if $$\text{Im } p_ i<0$$ for $$1\leq i\leq r$$, and $$\sigma(\overset \rightharpoonup\rho)=0$$ otherwise (here $${\overset \rightharpoonup\rho}:= (\rho_ 1,\dots, \rho_ r)$$). The author remarks that the precise definition of the exponential factor in the Weierstrass product formula for $$Z$$ can be best given “via zeta regularized determinant”, and he treats the case $$Z_ i(s)= 1-M_ i^{-s}$$, $$M_ i>1$$, $$1\leq i\leq r$$, in detail. This leads him to certain special functions $$F_ r(s)$$ expressible in terms of polylogarithms and, alternatively, in terms of the multiple gamma functions $$\Gamma_ r(z)$$ generalizing the double gamma function $$\Gamma_ 2(z)$$ of Barnes. Assuming $$Z_ i(s)$$, $$1\leq i\leq r$$, is a zeta-function defined by an Euler product, the author remarks that the multiple zeta-function $$Z(s)$$ would also have an Euler product decomposition, “at least formally”.
For the entire collection see [Zbl 0771.00036].
Reviewer: B.Z.Moroz (Bonn)

##### MSC:
 11M41 Other Dirichlet series and zeta functions 30D30 Meromorphic functions of one complex variable (general theory) 33B15 Gamma, beta and polygamma functions 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$