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arXiv · 1608.03014

Some sums over irreducible polynomials

David E Speyer
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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 $\mathbb{F}_q[T]$, 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 $\mathbb{F}_2[T]$, the sum $\sum_P \tfrac{1}{P^k - 1}$ always converges to a rational function, and is $0$ for $k=1$.

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David E Speyer. 2016-08-10. Some sums over irreducible polynomials. https://doi.org/10.2140/ant.2017.11.1231

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