Large Deviation Estimates of Selberg's Central Limit Theorem and Applications

2022/02/14 by Louis‐Pierre Arguin, Arguin, Louis-Pierre, Emma Bailey +1 · 4 citations
Mathematics · #11M06 #11M50 #60B20 #60G70 #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Number Theory (math.NT) #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2202.06799

openalex publication_date 2022/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For V∼ αloglog T with 0V\≪ (1)/(√(loglog T)) e^{-V2/loglog T}. \] This improves prior results of Soundararajan and of Harper on the large deviations of Selberg's Central Limit Theorem in that range, without the use of the Riemann hypothesis. The result implies the sharp upper bound for the fractional moments of the Riemann zeta function proved by Heap, Radziwiłł and Soundararajan. It also shows a new upper bound for the maximum of the zeta function on short intervals of length (log T)^θ, 0 0. Finally, it yields a sharp upper bound (to order one) for the moments on short intervals, below and above the freezing transition. The proof is an adaptation of the recursive scheme introduced by Bourgade, Radziwiłł and one of the authors to prove fine asymptotics for the maximum on intervals of length 1.

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• On the interim statistics for compact group characteristic polynomials and their derivatives
• Regulators of the fixed elliptic curve over rank-one imaginary quadratic fields
• Upper tail distributions of central L-values of quadratic twists of elliptic curves at the variance scale
• Selberg's Central Limit Theorem weighted by Linear Statistics of Zeta Zeros

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