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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)

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, \({}_r\phi_s\)