2021/09/21 by Hung M. Bui, Alexandra Florea, Bui, Hung M. +3
Computer Science · Mathematics · Social Sciences · #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Historical Geopolitical and Social Dynamics #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2109.10396
openalex publication_date 2021/09/21 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We prove special cases of the Ratios Conjecture for the family of quadratic\nDirichlet L--functions over function fields. More specifically, we study the\naverage of L(1/2+\α,\χD)/L(1/2+\β,\χD), when D varies over\nmonic, square-free polynomials of degree 2g+1 over mathbbFq[x], as g\n\→ \∞, and we obtain an asymptotic formula when Re \β \≫\ng-1/2+\ε. We also study averages of products of 2 over 2 and\n3 over 3 L--functions, and obtain asymptotic formulas when the shifts in\nthe denominator have real part bigger than g-1/4+\ε and\ng-1/6+\ε respectively. The main ingredient in the proof is\nobtaining upper bounds for negative moments of L--functions. The upper bounds\nwe obtain are expected to be almost sharp in the ranges described above. As an\napplication, we recover the asymptotic formula for the one-level density of\nzeros in the family with the support of the Fourier transform in (-2,2).\n