On the zeta functions on the projective complex spaces

الأصل · EN

In this article, we study the zeta function ζq associated to the Laplace operator Δq acting on the space of the smooth (0,q)-forms with q=0,,n on the complex projective space Pⁿ(C) endowed with its Fubini-Study metric. In particular, we show that the values of ζq at non-positive integers are rational. Moreover, we give a formula for ∑q≥ ₀(-1)q⁺¹qζq'(0), the associated holomorphic analytic torsion.