A Note on Zeta Measures over Function Fields

Let r be a power of a prime number p, Fr be the finite field of r elements, and Fr[T] be the polynomial ring over Fr. As an analogue to the Riemann zeta function over Z, Goss constructed the zeta function ζFr[T](s) over Fr[T]. In order to study this zeta function, Thakur calculated the divided power series associated to the zeta measure μx on Fr[T]v, where v is a finite place of Fr(T). This paper calculates the divided power series associated to the zeta measure on Fr[T]∞=Fr[[1T]] and expresses ζFr[T](s) by an integral of some locally analytic function.