科研之友
微信
新浪微博
Facebook
分享链接
摘要
Let k is an element of N* be even. We consider two trigonometric series F-k(x) = Sigma(infinity)(n = 1) sigma(k-1)(n)/n(k+1) sin(2 pi nx) and G(k)(x) = Sigma(infinity)(n = 1) sigma(k-1)(n)/n(k+1) cos(2 pi nx), where sigma(k-1) is the divisor function. They converge on 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.
- 出版日期2017-4
