Note on the a-points of the Riemann zeta function

2024/11/20 by Peng-Cheng Hang, Hang, Peng-Cheng, Min-Jie Luo +1
Mathematics · #11M06 #11M26 #41A60 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2411.13255

openalex publication_date 2024/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For any a∈ℂ, the zeros of ζ(s)-a, denoted by ρ_a=β_a+iγ_a, are called a-points of the Riemann zeta function ζ(s). In this paper, we reformulate some basic results about the a-points of ζ(s) shown by Garunkštis and Steuding. We then deduce an asymptotic of the sum S_T(a,δ)=∑_{τlt;γa\leqslant T}ζ'(ρ_a+iδ)X^ρ_a, T→∞, where 0≠δ=(2πα)/(log(T)/(2πX))≪ 1, and X>0 and τ\geqslant|δ|+1 are fixed. We also find the interesting varied behavior of S_T(a,δ) in different X ranges, which is more complicated than those described before by Gonek and Pearce-Crump.

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