On the zeta functions on the projective complex spaces

In this article, we study the zeta function $ζ_q$ associated to the Laplace operator $Δ_q$ 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 $ζ_q$ at non-positive integers are rational. Moreover, we give a formula for $\sum_{q\geq 0}(-1)^{q+1}qζ_q'(0),$ the associated holomorphic analytic torsion.

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