Some sums over irreducible polynomials

Speyer, David

doi:10.2140/ant.2017.11.1231

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Some sums over irreducible polynomials

David E. Speyer

Vol. 11 (2017), No. 5, 1231–1241

DOI: 10.2140/ant.2017.11.1231

Abstract

We prove a number of conjectures due to Dinesh Thakur concerning sums of the form $\sum_{P} h (P)$ where the sum is over monic irreducible polynomials $P$ in $F_{q} [T]$ , the function $h$ is a rational function and the sum is considered in the $T^{- 1}$ -adic topology. As an example of our results, in $F_{2} [T]$ , the sum $\sum_{P} 1 ∕ (P^{k} - 1)$ always converges to a rational function, and is $0$ for $k = 1$ .

Keywords

zeta function, special value, function field

Mathematical Subject Classification 2010

Primary: 11M38

Secondary: 05E05, 11M32

Milestones

Received: 17 October 2016

Accepted: 3 April 2017

Published: 12 July 2017

Authors

David E. Speyer
Department of Mathematics University of Michigan 2844 East Hall 530 Church Street Ann Arbor, MI 48109-1043 United States

Vol. 11, No. 5, 2017