Intermittency of Riemann's non-differentiable function through the fourth-order flatness

TL;DR

This paper investigates the intermittency of Riemann's non-differentiable function using fourth-order flatness, revealing its logarithmic divergence through multifractal analysis and dual space methods, linking mathematical properties to physical phenomena.

Contribution

It introduces a refined analysis of intermittency in Riemann's function via fourth-order flatness, combining physical and Fourier space approaches to demonstrate logarithmic divergence.

Findings

Fourth-order flatness diverges logarithmically.

Intermittency is characterized through multifractal analysis.

Dual approach confirms divergence in physical and Fourier spaces.

Abstract

Riemann's non-differentiable function is one of the most famous examples of continuous but nowhere differentiable functions, but it has also been shown to be relevant from a physical point of view. Indeed, it satisfies the Frisch-Parisi multifractal formalism, which establishes a relationship with turbulence and implies some intermittent nature. It also plays a surprising role as a physical trajectory in the evolution of regular polygonal vortices that follow the binormal flow. With this motivation, we focus on one more classic tool to measure intermittency, namely the fourth-order flatness, and we refine the results that can be deduced from the multifractal analysis to show that it diverges logarithmically. We approach the problem in two ways: with structure functions in the physical space and with high-pass filters in the Fourier space.