Zero-density estimates for L-functions attached to cusp forms

2013/10/02 by Yoshikatsu Yashiro, Yashiro, Yoshikatsu
Mathematics · #11M26 #11N75 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1310.0765

openalex publication_date 2013/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let S_k be the space of holomorphic cusp forms of weight k with respect to SL₂(ℤ). Let f ∈ S_k be a normalized Hecke eigenform, L_f(s) the L-function attached to the form f. In this paper we consider the distribution of zeros of L_f(s) in the strip σ≤ \Re s ≤ 1 for fixed σ>1/2 with respect to the imaginary part. We study estimates of N_f(σ,T) = #\ρ∈ℂ | L_f(ρ)=0, σ leq \Reρ≤ 1, 0 ≤ \Imρ≤ T for 1/2 ≤ σ≤1 and large T>0. Using the methods of Karatsuba and Voronin we shall give another proof for Ivić's method.

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