A note on zeta measures over function fields

Let r be a power of a prime number p, F-r be the finite field of r elements . and F-r[T] be the polynomial ring over F-r. As an analogue to the Riemann zeta function over Z. Goss constructed the zeta function zeta (Fr[T])(s) o ver F-r[T]. In order to study, this zeta function, Thakur calculated the di vided power series associated to the zeta measure mu (x) on F-r[ T](r), whe re r is a finite place of F-r( T). This paper calculates the divided power series associated to the Zeta measure on F-r[T](infinity) = F-r[[1/T]] and expresses zeta (Fr[T])(s) by an integral of some locally analytic function. (C) 2001 Academic Press.