Omega Theorems for Logarithmic Derivatives of Zeta and L-functions Near the 1-line

2024/04/26 by Zhonghua Li, Shengbo Zhao, Li, Zhonghua +1
Mathematics · #Mathematical functions and polynomials #Functional Equations Stability Results #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.2404.17250

Abstract

We establish an omega theorem for logarithmic derivative of the Riemann zeta function near the 1-line by resonance method. We show that the inequality | ζ^′(σ_A+it)/ζ(σ_A+it) | \geqslant ((e^A-1)/A)log₂ T + O(log₂ T / log₃ T) has a solution t ∈ [T^β, T] for all sufficiently large T, where σ_A = 1 - A / log₂ T.Furthermore, we give a conditional lower bound for the measure of the set of t for which the logarithmic derivative of the Riemann zeta function is large. Moreover, similar results can be generalized to Dirichlet L-functions.

Related

• Omega theorems for logarithmic derivatives of zeta and L-functions
• On derivatives of zeta and L-functions near the 1-line
• Omega-theorems for the Riemann zeta function and its derivatives near the line Re s=1
• Extreme values of the Riemann zeta function on the 1-line
• Bounding zeta on the 1-line under the partial Riemann hypothesis