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Determinants of Laplacians and multiple gamma functions. (English) Zbl 0641.33003

The author reinterpretes the classical formula ${\Gamma }\left(·\right)=\sqrt{\pi }$ in the form

${\Gamma }\left(·\right)={2}^{-1/2}{\left(det{{\Delta }}_{1}\right)}^{1/4},$

where ${{\Delta }}_{1}=-{d}^{2}/d{x}^{2}$ denotes the Laplacian on ${S}^{1}$. He then introduces so-called multiple Gamma functions ${{\Gamma }}_{n}$ for all $n\ge 0$ and then his main result states that ${{\Gamma }}_{n}\left(·\right)$ can be evaluated in terms of det ${{\Delta }}_{m}$ $\left(m=1,···,n\right)$, where ${{\Delta }}_{m}$ is the Laplacian on the m-sphere ${S}^{m}$. The proof splits into two parts: First, ${{\Gamma }}_{n}\left(·\right)$ is expressed in terms of the numbers ${\zeta }^{\text{'}}\left(-m\right)$ $\left(m=0,1,···,n-1\right)$, where $\zeta$ denotes the Riemann zeta function. Second, det ${{\Delta }}_{n}$ is also expressed in terms of ${\zeta }^{\text{'}}\left(-m\right)$ $\left(m=0,1,···,n-1\right)$. As a by-product, the author establishes the formula $logA=\left(1/12\right)-{\zeta }^{\text{'}}\left(-1\right)$ for the Kinkelin constant A.

The paper under review is closely related with work of A. Voros [Commun. Math. Phys. 110, 439-465 (1987; Zbl 0631.10025)] and 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

MSC:
 33B15 Gamma, beta and polygamma functions 58J50 Spectral problems; spectral geometry; scattering theory