An explicit form of Ingham's zero density estimate

2025/07/21 by Shashi Chourasiya, Chourasiya, Shashi, Aleksander Simonič +1
Mathematics · #11M06 #11M26 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2507.15184

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

Abstract

Ingham (1940) proved that N(σ,T)≪ T^{3(1-σ)/(2-σ)}log⁵T, where N(σ,T) counts the number of the non-trivial zeros ρ of the Riemann zeta-function with \Re\ρ\≥σ≥ 1/2 and 0<\Im\ρ\≤ T. We provide an explicit version of this result with the exponent (7-5σ)/(2-σ) of the logarithmic factor. In addition, we also provide an explicit estimate with asymptotically correct main term for the fourth power moment of the Riemann zeta-function on the critical line.

Citations

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Related

• Explicit zero density estimate for the Riemann zeta-function near the critical line
• An explicit version of Carlson's theorem
• An explicit log-free zero density estimate for the Riemann zeta-function
• A remark on density theorems for Riemann's zeta-function
• Almost all of the zeros of the Riemann zeta-function are on the critical line