## 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

### MSC:

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

### Citations:

Zbl 0631.10025; Zbl 0618.10023
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