The Ramanujan Journal ( IF 0.7 ) Pub Date : 2021-05-05 , DOI: 10.1007/s11139-021-00422-x J. C. Andrade , H. Jung , A. Shamesaldeen
In this paper, we extend to the function field setting the heuristics formerly developed by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments of L-functions. We also adapt to the function field setting the heuristics first developed by Conrey, Farmer and Zirnbauer to the study of mean values of ratios of L-functions. Specifically, the focus of this paper is on the family of quadratic Dirichlet L-functions \(L(s,\chi _{P})\) where the character \(\chi \) is defined by the Legendre symbol for polynomials in \(\mathbb {F}_{q}[T]\) with \(\mathbb {F}_{q}\) a finite field of odd cardinality, and the averages are taken over all monic and irreducible polynomials P of a given odd degree. As an application, we also compute the formula for the one-level density for the zeros of these L-functions.
中文翻译:
$$ \ mathbb {F} _ {q} [T] $$ F q [T]中的一元不可约多项式的二次Dirichlet L-函数的积分矩和比率
在本文中,我们扩展到函数域,以设置先前由Conrey,Farmer,Keating,Rubinstein和Snaith开发的启发式方法用于L函数的积分矩。我们还适应了函数领域,设置了由Conrey,Farmer和Zirnbauer最初开发的启发式方法,以研究L函数比率的平均值。具体来说,本文的重点是二次Dirichlet L函数\(L(s,\ chi _ {P})\)的族,其中字符\(\ chi \)由Legendre符号定义为多项式\(\ mathbb {F} _ {q} [T] \)与\(\ mathbb {F} _ {q} \)奇数基数的有限域,并且平均值取于给定奇数度的所有一元和不可约多项式P。作为应用程序,我们还计算了这些L函数的零点的单级密度的公式。