## Algebra & Number Theory

- Algebra Number Theory
- Volume 11, Number 5 (2017), 1231-1241.

### Some sums over irreducible polynomials

#### 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 ${\sum}_{P}1\u2215\left({P}^{k}-1\right)$ always converges to a rational function, and is $0$ for $k=1$.

#### Article information

**Source**

Algebra Number Theory, Volume 11, Number 5 (2017), 1231-1241.

**Dates**

Received: 17 October 2016

Accepted: 3 April 2017

First available in Project Euclid: 12 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ant/1513090726

**Digital Object Identifier**

doi:10.2140/ant.2017.11.1231

**Mathematical Reviews number (MathSciNet)**

MR3671435

**Zentralblatt MATH identifier**

06748170

**Subjects**

Primary: 11M38: Zeta and $L$-functions in characteristic $p$

Secondary: 05E05: Symmetric functions and generalizations 11M32: Multiple Dirichlet series and zeta functions and multizeta values

**Keywords**

zeta function special value function field

#### Citation

Speyer, David. Some sums over irreducible polynomials. Algebra Number Theory 11 (2017), no. 5, 1231--1241. doi:10.2140/ant.2017.11.1231. https://projecteuclid.org/euclid.ant/1513090726