On the zeta functions on the projective complex spaces

In this article, we study the zeta function ${\zeta }_q$ associated to the Laplace operator ${\mathrm{\Delta }}_q$ acting on the space of the smooth $(0,q)$-forms with $q=0,\dots ,n$ 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 ${\sum }_{q\ge 0}(-1{)}^{q+1}q{\zeta }_q^{\mathrm{\prime }}(0),$ the associated holomorphic analytic torsion.