Generalised heat coefficients and associated spectral zeta functions on complex projective spaces Pn(ℂ)

Abstract

ABSTRACTThe heat invariants or the Minakshisundaram-Pleijel heat coefficients describe the residues of the spectral zeta functions on any compact Riemannian manifold M. In this paper, we first give an explicit description of the heat coefficients via the Jacobi theta functions and their higher order derivatives, and then express these Minakshisundaram-Pleijel coefficients in terms of the residues of the associated zeta functions . Different interesting formulae are obtained for these zeta functions. A new class of heat coefficients, the Maclaurin heat coefficients () (i.e. the coefficients appearing in the Maclaurin expansion of the heat kernel ), are explicitly studied in terms of the classical and generalised Minakshisundaram-Pleijel coefficients and () respectively. Remarkable asymptotic expansions for the Maclaurin spectral functions are established. We also introduce and construct new zeta functions associated with these Maclaurin heat coefficients (generalised Minakshisundaram-Pleijel zeta functions), and it is interesting to see that these generalised zeta functions are explicitly understood in terms of the classical (Minakshisundaram-Pleijel) zeta functions.