Let $\underline{ \mathcal F}$ be a $\tau$-sheaf. Building on previous work of Drinfeld, Anderson, Taguchi, and Wan, B\"ockle and Pink \cite{ bp1} develop a cohomology theory for $\underline{ \mathcal F}$. In \cite{ boc1} B\"ockle uses this theory to establish the analytic continuation of the $L$-series associated to $\underline{ \mathcal F}$ (which is a characteristic $p$ valued ``Dirichlet series'') { \em and} the logarithmic growth of the degrees of its special polynomials. In this paper we shall show that this logarithmic growth is all that is needed to analytically continue the original $L$-series as well as { \em all} associated partial $L$-series. Moreover, we show that the degrees of the special polynomials attached to the partial $L$-series also grow logarithmically. Our tools are B\"ockle's original results, non-Archimedean integration, and the very strong estimates of Y. Amice \cite{ am1}. Along the way, we define certain natural modules associated with non-Archimedean measures (in the characteristic 0 case as well as in characteristic $p$).
Categories: math.NT