ZETA-REGULARIZED PRODUCTS

If lambda(k) is a sequence Of nonzero complex numbers, then we define the zeta regularized product of these numbers, PI(k)lambda(k), to be exp(-Z'(0)) where Z(s) = SIGMA(k=0)(infinity)lambda(k)-s. We assume that Z(s) has analytic continuation to a neighborhood of the origin. If lambda(k) is the sequence of positive eigenvalues of the Laplacian on a manifold, then the zeta regularized product is known as det'DELTA, the determinant of the Laplacian, and PI(k)(lambda(k) - lambda) is known as the functional determinant. The purpose of this paper is to discuss properties of the determinant and functional determinant for general sequences of complex numbers. We discuss asymptotic expansions of the functional determinant as lambda --> -infinity and its relationship to the Weierstrass product. We give some applications to the theory of Barnes' multiple gamma functions and elliptic functions. A new proof is given for Kronecker's limit formula and the product expansion for Barnes' double Stirling modular constant.