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

The author reinterpretes the classical formula \(\Gamma(.)=\sqrt{\pi}\) in the form \[ \Gamma(.)=2^{-1/2}(\det \Delta_ 1)^{1/4}, \] where \(\Delta_ 1=-d^ 2/dx^ 2\) denotes the Laplacian on \(S^ 1\). He then introduces so-called multiple Gamma functions \(\Gamma_ n\) for all \(n\geq 0\) and then his main result states that \(\Gamma_ n(.)\) can be evaluated in terms of det \(\Delta_ m\) \((m=1,...,n)\), where \(\Delta_ m\) is the Laplacian on the m-sphere \(S^ m\). The proof splits into two parts: First, \(\Gamma_ n(.)\) is expressed in terms of the numbers \(\zeta'(-m)\) \((m=0,1,...,n-1)\), where \(\zeta\) denotes the Riemann zeta function. Second, det \(\Delta_ n\) 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 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


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