On the zeta functions on the projective complex spaces

Abstract: In this article, we study the zeta function

z e t a_{q}

associated to the Laplace operator

D e l t a_{q}

acting on the space of the smooth

(0, q)

-forms with

q = 0, l d o t s, n

on the complex projective space

m a t h b b P^{n} (m a t h b b C)

endowed with its Fubini-Study metric. In particular, we show that the values of

z e t a_{q}

at non-positive integers are rational. Moreover, we give a formula for

s u m_{q g e q 0} (- 1)^{q + 1} q z e t {a_{q}}^{'} (0),

the associated holomorphic analytic torsion.