Differentiability of arithmetic Fourier series arising from Eisenstein series

الأصل · EN

Let k be even. We consider two series Fₖ(x)= ∑ₙ₌₁∞ σₖ₋₁(n)nᵏ⁺¹ (2πn x) and Gₖ(x)= ∑ₙ₌₁∞ σₖ₋₁(n)nᵏ⁺¹ (2πn x), where σₖ₋₁ is the divisor function. They converge on R to continuous functions. In this paper, we examine the differentiability of Fₖ and Gₖ. 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 F₂ at an irrational x is related to the fine diophantine properties of x. We estimate the modulus of continuity of F₂. We formulate a conjecture concerning differentiability of Fₖ and Gₖ for any k even.