# Limit Theorems for a class of unbounded observables with an application to "Sampling the Lindel\"of hypothesis"

## TL;DR

This paper establishes limit theorems such as the CLT and Edgeworth Expansion for unbounded oscillating observables over expanding maps, with applications to the Riemann zeta function and the Lindelöf hypothesis.

## Contribution

It provides the first CLT, Edgeworth Expansion, and MLCLT for a class of unbounded oscillating functions, including parts of the Riemann zeta function, over expanding maps.

## Key findings

- Proved CLT, Edgeworth Expansion, and MLCLT for unbounded oscillating observables.
- Extended results to Boolean-type transformations on the real line.
- Applied the theorems to the real and imaginary parts of the Riemann zeta function.

## 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 $C^2$ expanding maps of the interval. As a corollary, we obtain the corresponding results for Boolean-type transformations on $\mathbb{R}$. The class of observables in the CLT and the MLCLT on $\mathbb{R}$ 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 (2009) and Steuding (2012) who have proven the Strong Law of Large Numbers for "Sampling the Lindel\"of hypothesis".