On the zeta functions on the projective complex spaces

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.