On the zeta functions on the projective complex spaces

TL;DR

This paper investigates the zeta functions related to Laplace operators on complex projective spaces, revealing rationality of their values at non-positive integers and deriving a formula for the associated holomorphic analytic torsion.

Contribution

It provides new results on the rationality of zeta function values and explicit formulas for holomorphic analytic torsion on complex projective spaces.

Findings

Values of $z_q$ at non-positive integers are rational.

Derived a formula for the holomorphic analytic torsion.

Enhanced understanding of spectral invariants on complex projective spaces.

Abstract

In this article, we study the zeta function $\zeta_q$ associated to the Laplace operator $\Delta_q$ acting on the space of the smooth $(0,q)$-forms with $q=0,\ldots,n$ on the complex projective space $\mathbb{P}^n(\mathbb{C})$ endowed with its Fubini-Study metric. In particular, we show that the values of $\zeta_q$ at non-positive integers are rational. Moreover, we give a formula for $ \sum_{q\geq 0}(-1)^{q+1}q\zeta_q'(0),$ the associated holomorphic analytic torsion.