Differentiability of arithmetic Fourier series arising from Eisenstein series

TL;DR

This paper investigates the differentiability of specific Fourier series linked to Eisenstein series, revealing that their smoothness depends on number-theoretic properties of the points, especially for the case k=2.

Contribution

It establishes the differentiability criteria for these series at irrational points and connects it to Diophantine approximation, providing new insights into their regularity.

Findings

Differentiability of F_2 at irrationals depends on Diophantine properties.

The sine series F_2 behaves differently from the cosine series G_2 in terms of smoothness.

A conjecture is proposed for the differentiability of F_k and G_k for all even k.

Abstract

Let $k$ be even. We consider two series $F_k(x)= \sum_{n=1}^\infty \frac{\sigma_{k-1}(n)}{n^{k+1}} \sin(2\pi n x)$ and $G_k(x)= \sum_{n=1}^\infty \frac{\sigma_{k-1}(n)}{n^{k+1}} \cos(2\pi n x)$, where $\sigma_{k-1}$ is the divisor function. They converge on $\mathbb{R}$ to continuous functions. In this paper, we examine the differentiability of $F_k$ and $G_k$. These functions are related to Eisenstein series and their (quasi-)modular properties allow us to apply the method proposed by Itatsu in 1981 in the study of the Riemann series. We focus on the case $k=2$ and we show that the sine series exhibits different behaviour with respect to differentiability than the cosine series. We prove that the differentiability of $F_2$ at an irrational $x$ is related to the fine diophantine properties of $x$. We estimate the modulus of continuity of $F_2$. We formulate a conjecture concerning differentiability of $F_k$ and $G_k$ for any $k$ even.