VALUATIONS OF V-ADIC POWER SUMS AND ZERO DISTRIBUTION FOR GOSS’ V-ADIC ZETA FOR Fq[t]

We study the valuation at an irreducible polynomial v of the vadic power sum, for exponent k (or −k), of polynomials of a given degree d in Fq [t], as a sequence in d (or k). Understanding these sequences has immediate consequences, via standard Newton polygon calculations, for the zero distribution of corresponding v-adic Goss zeta functions. We concentrate on v of degree one and two and give several results and conjectures describing these sequences. As an application, we show, for example, that the naive Riemann hypothesis statement which works in several cases, needs modifications, even for a prime of degree two. In the appendix, we give an elementary proof of (and generalize) a product formula of Richard Pink for the leading term of the Goss zeta function.