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

The author reinterpretes the classical formula Γ(·)=π in the form

Γ(·)=2 -1/2 (detΔ 1 ) 1/4 ,

where Δ 1 =-d 2 /dx 2 denotes the Laplacian on S 1 . He then introduces so-called multiple Gamma functions Γ n for all n0 and then his main result states that Γ n (·) can be evaluated in terms of det Δ m (m=1,···,n), where Δ m is the Laplacian on the m-sphere S m . The proof splits into two parts: First, Γ n (·) is expressed in terms of the numbers ζ ' (-m) (m=0,1,···,n-1), where ζ denotes the Riemann zeta function. Second, det Δ n is also expressed in terms of ζ ' (-m) (m=0,1,···,n-1). As a by-product, the author establishes the formula logA=(1/12)-ζ ' (-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:
33B15Gamma, beta and polygamma functions
58J50Spectral problems; spectral geometry; scattering theory