Mod-$\phi$ convergence

TL;DR

This paper explores the use of Fourier analysis to establish local limit theorems in weak convergence, with applications spanning random matrix theory, number theory, complex Brownian motion, and the Riemann zeta function.

Contribution

It introduces a framework based on mod-$oldsymbol{ extphi}$ convergence for analyzing local limit theorems across various probabilistic and number-theoretic contexts.

Findings

Established local limit theorems in diverse models

Connected Fourier analysis with mod-$oldsymbol{ extphi}$ convergence

Provided insights into the distribution of Riemann zeta function values

Abstract

Using Fourier analysis, we study local limit theorems in weak-convergence problems. Among many applications, we discuss random matrix theory, some probabilistic models in number theory, the winding number of complex brownian motion and the classical situation of the central limit theorem, and a conjecture concerning the distribution of values of the Riemann zeta function on the critical line.