The Developmental Zeta Function

The developmental zeta function is the spectral zeta function of the developmental operator D = nabla-star nabla, the Bochner Laplacian on a compact developmental manifold (M, gD) of Developmental Geometry (Book 11). Using the discrete spectrum and heat semigroup established in Book 11 (Theorems 1. 12 and 1. 13), we define zetaD (s) as the Dirichlet series of inverse eigenvalues summed to the power -s, prove absolute convergence in the half-plane Re (s) > dim (M) /2 via the Weyl law, and establish meromorphic continuation to the complex plane with simple poles at s = dim (M) /2 - k for k a non-negative integer via the Mellin transform and the Seeley-DeWitt heat kernel expansion. The residues are the Seeley-DeWitt coefficients determined by the geometry of (M, gD). In dimension 2, there is a simple pole at s = 1 with residue vol (M, gD) / (2 pi) ; zetaD is regular at s = 0 and at negative integers. A functional equation and zero-free region are flagged as open directions. The present paper establishes the analytic foundation on which such questions can be posed rigorously.