The author reinterpretes the classical formula in the form
where denotes the Laplacian on . He then introduces so-called multiple Gamma functions for all and then his main result states that can be evaluated in terms of det , where is the Laplacian on the m-sphere . The proof splits into two parts: First, is expressed in terms of the numbers , where denotes the Riemann zeta function. Second, det is also expressed in terms of . As a by-product, the author establishes the formula 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.