Moments of Higher Logarithmic Derivatives and Exceptional Zeros in Cyclotomic Fields

2025/09/08 by Samprit Ghosh, Ghosh, Samprit
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Mathematical Approximation and Integration #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2509.06390

openalex publication_date 2025/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let χ be a non-principal Dirichlet character, and let L(s,χ) be the associated Dirichlet L-function. We write L(s,χ) for its logarithmic derivative L'(s,χ)/L(s,χ). In this article, we prove arithmetic formulas for the higher derivatives L^(r)(1,χ) and establish unconditional moment asymptotics for P^(a,b)(L^(r)(1,χ)) as χ runs over non-principal Dirichlet characters of large prime conductor. As an application, we show that these moment asymptotics imply positivity of the second Li coefficient of prime cyclotomic fields of sufficiently large conductor. Combined with a zero-free criterion in terms of this Li coefficient, this rules out the possible exceptional zero in a Stark-type zero-free region for these fields.

Citations

• Higher Euler-Kronecker Constants of Number fields
• On the zeros of L-functions
• Fractional Moments of Dirichlet L-Functions
• Moments of the Riemann zeta-function
• Probability laws related to the Jacobi theta and Riemann zeta function and Brownian excursions
• The Positivity of a Sequence of Numbers and the Riemann Hypothesis

Related

• The eighth moment of Dirichlet L-functions II
• On moments of L-functions over Dirichlet characters
• The Fourth Moment of Derivatives of Dirichlet L-functions in Function Fields
• On the Moments of Certain Families of Dirichlet L-functions
• Moments of Dirichlet L-functions with prime conductors over function\n fields