On the number of zeros of \mathop\mathcal R(s)

2024/06/13 by Juan Arias de Reyna, de Reyna, Juan Arias
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical Approximation and Integration #Number Theory (math.NT) #Primary 11M06 #Secondary 30D99

paper · pdf · doi:10.48550/arxiv.2406.08890

openalex publication_date 2024/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the number of zeros \varrho=β+iγ of \mathop\mathcal R(s) with 0<γ≤ T is given by N(T)=(T)/(4π)log(T)/(2π)-(T)/(4π)-\frac12√((T)/(2π))+O(T^2/5log² T). Here \mathop\mathcal R(s) is the function that Siegel found in Riemann's papers. Siegel related the zeros of \mathop\mathcal R(s) to the zeros of Riemann's zeta function. Our result on N(T) improves the result of Siegel.

Related

• Report on some papers related to the function \mathop\mathcal R (s) found by Siegel in Riemann's posthumous papers
• The Number of Zeros of ζ'(s)
• On Siegel results about the zeros of the auxiliary function of Riemann
• Counting zeros of the Riemann zeta function
• Statistic of zeros of Riemann auxiliary function