Irreducible polynomials in F_q[x] versus 1 - qz in C[z]
Number Theory
2007-05-23 v3
Abstract
Let q be a prime power and N_n count degree-n monic irreducible polynomials in F_q[x]. We show that prod_{n=1}^{\infty} (1 -z^n)^N_n = 1 - qz for small |z|.
Cite
@article{arxiv.math/0612614,
title = {Irreducible polynomials in F_q[x] versus 1 - qz in C[z]},
author = {Barry Brent},
journal= {arXiv preprint arXiv:math/0612614},
year = {2007}
}
Comments
2 pages. v.1 replaced because the proof of absolute convergence was valid only on reals. (It used l'Hopital.) Sadder news: "...you've simply rediscovered the Weil conjectures for P^1. ..." (email from Kevin Buzzard)