The Rate of Convergence for Selberg's Central Limit Theorem under the Riemann Hypothesis

2023/08/18 by Roberts, Asher
#11M06 #60F05 #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.2308.09679

Abstract

We assume the Riemann hypothesis to improve upon the rate of convergence of (logloglog T)²/√(loglog T) in Selberg's central limit theorem for log|ζ(1/2+it)| given by the author. We achieve a rate of convergence of √(loglogloglog T)/√(loglog T) in the Dudley distance. The proof is an adaptation of the techniques used by the author, based on the work of Radziwill and Soundararajan and Arguin et al., combined with a lemma of Selberg that provides for a mollifier close to the critical line Re(s)=1/2 under the Riemann hypothesis.

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