A discrete mean value of the Riemann zeta function

2023/11/22 by Kübra Benli̇, Benli, Kübra, Ertan Elma +3 · 3 citations
Mathematics · #11M26 #Analytic Number Theory Research #Analytic and geometric function theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT) #Primary 11M06 #Secondary 11N37

paper · pdf · doi:10.48550/arxiv.2311.13554

openalex publication_date 2023/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this work, we estimate the sum ∑_{0 lt; \Im(ρ) ≤ T} ζ(ρ+α)X(ρ) Y(1 - ρ) over the nontirival zeros ρ of the Riemann zeta funtion where α is a complex number with α≪ 1/log T and X(⋅) and Y(⋅) are some Dirichlet polynomials. Moreover, we estimate the discrete mean value above for higher derivatives where ζ(ρ+α) is replaced by ζ^(m)(ρ) for all m∈ℕ. The formulae we obtain generalize a number of previous results in the literature. As an application, assuming the Riemann Hypothesis we obtain the lower bound ∑_{0 lt; \Im(ρ) lt; T} | ζ^(m)(ρ)|^2k ≫ T(log T)^k2+2km+1 (k,m∈ℕ) which was previously known under the Generalized Riemann Hypothesis, in the case m=1.

Cited by

• Mean values of the Riemann zeta function at shifted zeros under the Riemann Hypothesis
• Complex moments of the derivative of the Riemann zeta function
• Prime Number Error Terms

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