On the regularity of fractional integrals of modular forms

TL;DR

This paper investigates the regularity, differentiability, and singularity spectrum of fractional integrals of modular forms, providing new insights into their local and global properties and their functional equations.

Contribution

It characterizes the differentiability at rational points, computes the Hölder spectrum, and analyzes the graphs of fractional integrals of modular forms, including cusp forms.

Findings

Characterized differentiability at rational points.

Determined the Hölder exponent everywhere.

Computed the spectrum of singularities.

Abstract

In this paper we study some local and global regularity properties of Fourier series obtained as fractional integrals of modular forms. In particular we characterize the differentiability at rational points, determine their H\"older exponent everywhere (using several definitions) and compute the associated spectrum of singularities. We also show that these functions satisfy an approximate functional equation, and use it to discuss the graphs of "Riemann's example" and of fractional integrals of cusp forms for $\Gamma_0(N)$. We include some computer plots.