F_q[x] 中的不可约多项式与 C[z] 中的 1 - qz
数论
2007-05-23 v3
摘要
设 q 为素数幂，N_n 计数 F_q[x] 中 n 次首一不可约多项式的个数。我们证明对于小的 |z|，有 prod_{n=1}^{\infty} (1 -z^n)^N_n = 1 - qz。
引用
@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}
}
备注
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)