Zeros of higher derivatives of Riemann zeta function

2021/04/20 by Mithun Das, Das, Mithun Kumar, Sudhir Pujahari +1
Mathematics · #11M #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2104.10243

openalex publication_date 2021/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, we extend the result of Conrey [5, Theorem 2] to shorter intervals for higher-order derivatives of the zeta function. That is we study the mean value of the product of two finite order derivatives of the zeta function multiplied by a mollifier in short intervals. In this process, we obtain better mollifier length in some short intervals compared to the length of mollifier implied by Conrey's result. These finer studies allow us to refine the error term of some classical results of Levinson and Montgomery [13], Ki and Lee [11] on zero density estimates of ζ^(k). Further, we showed that almost all non-trivial zeros of Matsumoto-Tanigawa's η_k-function cluster near the critical line.

Related

• Zeros of high derivatives of the Riemann Zeta function
• Distribution of the zeros of the Riemann zeta function in longer intervals
• Explicit zero density estimate for the Riemann zeta-function near the critical line
• Almost all of the zeros of the Riemann zeta-function are on the critical line
• More than five-twelfths of the zeros of ζ are on the critical line