Differentiability of arithmetic Fourier series arising from Eisenstein series

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.