Vol. 11, No. 5, 2017

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Some sums over irreducible polynomials

David E. Speyer

Vol. 11 (2017), No. 5, 1231–1241

We prove a number of conjectures due to Dinesh Thakur concerning sums of the form Ph(P) where the sum is over monic irreducible polynomials P in Fq[T], the function h is a rational function and the sum is considered in the T1-adic topology. As an example of our results, in F2[T], the sum P1(Pk 1) always converges to a rational function, and is 0 for k = 1.

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zeta function, special value, function field
Mathematical Subject Classification 2010
Primary: 11M38
Secondary: 05E05, 11M32
Received: 17 October 2016
Accepted: 3 April 2017
Published: 12 July 2017
David E. Speyer
Department of Mathematics
University of Michigan
2844 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States