Differentiability of arithmetic Fourier series arising from Eisenstein series

Let (kin mathbb {N}^*) be even. We consider two trigonometric series ( F_k(x)= sum _{n=1}^infty frac{sigma _{k-1}(n)}{n^{k+1}} sin (2pi n x)) and (G_k(x)= sum _{n=1}^infty frac{sigma _{k-1}(n)}{n^{k+1}} cos (2pi 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 a 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 continued fraction expansion 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.