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Valuations of ***v*-adic Power Sums and Zero Distribution for the
Goss *v*-adic Zeta Function for **F**_{q}[*t*]

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Dinesh S. Thakur

Department of Mathematics

University of Arizona

Tucson, AZ 85721

USA

**Abstract:**

We study the valuation at an irreducible polynomial *v* of the
*v*-adic power sum, for exponent *k* (or -*k*),
of polynomials of a
given degree *d* in **F**_{q}[*t*],
as a sequence in *d* (or *k*).
Understanding these sequences has immediate consequences, via
standard Newton polygon calculations, for the zero distribution of the
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 last
section, we give an elementary proof of (and generalize) a product
formula of Pink for the leading term of the Goss zeta function.

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Received August 1 2012;
revised version received January 7 2013.
Published in *Journal of Integer Sequences*, March 2 2013.
Minor revisions, March 13 2014.

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