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Determinants of Laplacians and multiple gamma functions. (English) Zbl 0641.33003
The author reinterpretes the classical formula $\Gamma(.)=\sqrt{\pi}$ in the form $$ \Gamma(.)=2\sp{-1/2}(\det \Delta\sb 1)\sp{1/4},$$ where $\Delta\sb 1=-d\sp 2/dx\sp 2$ denotes the Laplacian on $S\sp 1$. He then introduces so-called multiple Gamma functions $\Gamma\sb n$ for all $n\ge 0$ and then his main result states that $\Gamma\sb n(.)$ can be evaluated in terms of det $\Delta\sb m$ $(m=1,...,n)$, where $\Delta\sb m$ is the Laplacian on the m-sphere $S\sp m$. The proof splits into two parts: First, $\Gamma\sb n(.)$ is expressed in terms of the numbers $\zeta'(-m)$ $(m=0,1,...,n-1)$, where $\zeta$ denotes the Riemann zeta function. Second, det $\Delta\sb n$ is also expressed in terms of $\zeta'(-m)$ $(m=0,1,...,n-1)$. As a by-product, the author establishes the formula $\log A=(1/12)-\zeta'(-1)$ for the Kinkelin constant A. The paper under review is closely related with work of {\it A. Voros} [Commun. Math. Phys. 110, 439-465 (1987; Zbl 0631.10025)] and {\it P. Sarnak} [Commun. Math. Phys. 110, 113-120 (1987; Zbl 0618.10023)]. In particular, Voros points out that A already was computed in the literature.
Reviewer: J.Elstrodt

33B15Gamma, beta and polygamma functions
58J50Spectral problems; spectral geometry; scattering theory
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