A Central Limit Theorem for Linear Combinations of Logarithms of\n Dirichlet L-functions

2021/09/19 by Fatma Çi̇çek, Fatma Çiçek, Çiçek, Fatma
Mathematics · #11M06 #11M26 #Advanced Mathematical Identities #Analytic Number Theory Research #Analytic and geometric function theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2109.09097

openalex publication_date 2021/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The purpose of this paper is to generalize our earlier work on the logarithm\nof the Riemann zeta-function to linear combinations of logarithms of primitive\nDirichlet L-functions with constant real coefficients. Under the assumption\nof suitable hypotheses, we prove that as T\→ \∞ , a sequence of the form\n(a₁\log|L(\ρ,\χ_n)|+\…+a₁\log|L(\ρ,\χ_n)|) has an approximate\nGaussian distribution with mean 0 and variance \n tfrac12\(a₁²+\…+a_n² \)\log\log T. Here a₁, \…, a_n\n\∈ \ℝ, each of the \χ_i is a primitive Dirichlet character modulo\nM_i with M_i\≤ T, and 0< \Im\ρ \≤ T where \ρ runs\nover nontrivial zeros of the zeta-function. From the proof of this result, we\nalso derive the independence of the distributions of sequences\n(\log|L(\ρ,\χ₁)|), \…, (\log|L(\ρ,\χ_n)|) provided that they are\nsuitably normalized.\n

Related

• Weighted central limit theorems for central values of L-functions
• Selberg's Central limit theorem for quadratic Dirichlet L-functions over function fields
• Simultaneous non-vanishing of Dirichlet L--functions, II: Weighted central limit theorem
• Linear Combinations of Logarithms of L-functions over Function Fields at Microscopic Shifts and Beyond
• Upper Bounds on Large Deviations of Dirichlet L-functions in the q-aspect