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August 17, 2025

Limit theorems for a class of unbounded observables with an application to ‘Sampling the Lindelöf hypothesis’

Key Points

The central limit theorem shows established limit behavior for unbounded observables under specific conditions.
Findings include the applicability of Birkhoff sums to the Riemann zeta function and its properties.
The approach relies on piecewise expanding maps, which are critical in determining observable behavior.
This work supports further understanding of the Lindelöf hypothesis and its sampling through various transformations.

Abstract

Abstract We prove the central limit theorem (CLT), the first-order Edgeworth expansion and a mixing local central limit theorem (MLCLT) for Birkhoff sums of a class of unbounded heavily oscillating observables over a family of full-branch piecewise C² expanding maps of the interval. As a corollary, we obtain the corresponding results for Boolean-type transformations on R. The class of observables in the CLT and the MLCLT on R include the real part, the imaginary part and the absolute value of the Riemann zeta function. Thus obtained CLT and MLCLT for the Riemann zeta function are in the spirit of the results of Lifschitz & Weber Sampling the Lindelöf hypothesis with the Cauchy random walk. Proc. Lond. Math. Soc. (3) 98 (2009), 241–270 and Steuding Sampling the Lindelöf hypothesis with an ergodic transformation. RIMS Kôkyûroku Bessatsu B34 (2012), 361–381 who have proven the strong law of large numbers for sampling the Lindelöf hypothesis.

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