Fourier series concerning Gibbs constant and the divisor function.

Question

It is quite a remarkable function I found. It seems, though, that I may be staring at something trivial, which is hopefully not the case. I would like some opinions. The function is $$f(b)=\lim\limits_{m\to\infty}\frac{2}{\pi}\sum_{n=1}^m\frac{\sin\left(2\pi n\left(\frac{x}{b}+\frac{1}{2m+2}\right)\right)-\sin\left(2\pi n\frac{x}{b}\right)}{n}=\begin{cases} G & \text{for $b|x$} \\ 0 & \text{for $b\nmid x$} \end{cases}$$ Where $G$ is the Gibbs constant. From it we can formulate a kind of Fourier transform for the divisor function. $$\sigma_a(x)=\lim\limits_{m\to\infty}\frac{1}{G}\sum_{b=1}^x\left(b^af(b)\right)$$ I'm wondering why I haven't seen $f(b)$ or variations used anywhere else. It seems incredibly useful to me. I know that many Fourier series such as that for the mod function (from which this was derived) does not converge to the true value at the points of discontinuity. However $f(b)$ does not run into such a problem. I think it has something to do with the Gibbs phenomenon (which I know very little about). Nevertheless this does seem quite phenomenal to me. I am just a high school student so I would very much appreciate some enlightened insight as to whether or not this is special. Also is there a neater or simpler way to express $f(b)$?