#### Vol. 11, No. 5, 2017

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Some sums over irreducible polynomials

### David E. Speyer

Vol. 11 (2017), No. 5, 1231–1241
##### Abstract

We prove a number of conjectures due to Dinesh Thakur concerning sums of the form ${\sum }_{P}h\left(P\right)$ where the sum is over monic irreducible polynomials $P$ in ${\mathbb{F}}_{q}\left[T\right]$, 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 ${\mathbb{F}}_{2}\left[T\right]$, the sum ${\sum }_{P}1∕\left({P}^{k}-1\right)$ 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