2017/01/05 by Sunghan Bae, Bae, Sunghan, Hwanyup Jung +1
Computer Science · Mathematics · #11R11 #11R29 #11R42 #11R58 #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1701.01493
openalex publication_date 2017/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k = \mathbbFq(T) be the rational function field over a finite field \mathbbFq, where q is a power of 2. In this paper we solve the problem of averaging the quadratic L-functions L(s, χu) over fundamental discriminants. Any separable quadratic extension K of k is of the form K = k(xu), where xu is a zero of X2+X+u=0 for some u∈ k. We characterize the family \mathcal I (resp. \mathcal F, \mathcal F') of rational functions u∈ k such that any separable quadratic extension K of k in which the infinite prime ∞ = (1/T) of k ramifies (resp. splits, is inert) can be written as K = k(xu) with a unique u∈\mathcal I (resp. u∈\mathcal F, u∈\mathcal F'). For almost all s∈\mathbb C with \rm Re(s)≥ \frac12, we obtain the asymptotic formulas for the summation of L(s,χu) over all k(xu) with u∈ \mathcal I, all k(xu) with u∈ \mathcal F or all k(xu) with u∈ \mathcal F' of given genus. As applications, we obtain the asymptotic mean value formulas of L-functions at s=\frac12 and s=1 and the asymptotic mean value formulas of the class number hu or the class number times regulator hu Ru.