Density theorems for Riemann's auxiliary function

2024/06/21 by Juan Arias de Reyna, de Reyna, Juan Arias
Mathematics · #Analytic and geometric function theory #FOS: Mathematics #Functional Equations Stability Results #Meromorphic and Entire Functions #Number Theory (math.NT) #Primary 11M06 #Secondary 30D99

paper · pdf · doi:10.48550/arxiv.2406.14987

openalex publication_date 2024/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a density theorem for the auxiliar function \mathop\mathcal R(s) found by Siegel in Riemann papers. Let α be a real number with \frac12< α≤ 1, and let N(α,T) be the number of zeros ρ=β+iγ of \mathop\mathcal R(s) with 1≥ β≥α and 0<γ≤ T. Then we prove N(α,T)≪ T^\frac32-α(log T)³. Therefore, most of the zeros of \mathop\mathcal R(s) are near the critical line or to the left of that line. The imaginary line for π^-s/2Γ(s/2)\mathop\mathcal R(s) passing through a zero of \mathop\mathcal R(s) near the critical line frequently will cut the critical line, producing two zeros of ζ(s) in the critical line.

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