In this answer it is explained that a big reason that the nth prime formula discussed with (13) and (14) isn't "useful" is because $\lfloor\frac{x}{b}\rfloor-\lfloor\frac{x-1}{b}\rfloor$ doesn't have "nice analytical properties". This special floor function is $1$ when $x$ divides $b$ and $0$ otherwise. I'm curious, does $$f(x)=\lim\limits_{m\to\infty}\frac{1}{m}\sum_{n=1}^m \cos\left(\frac{2\pi nx}{b}\right)$$ have good analytical properties? In other words, is it "useful"? It essentially performs the same function as the one referenced.
EDIT (Feb 12):
I've found that there is another equivalent expression we can write where, as @user1952009 noted there is for the above expression, there is no inversion of limits in this expression. Observe $$f(x)=\lim\limits_{m\to\infty}\frac{1}{2^m}\sum_{n=1}^{2^m} \cos\left(\frac{2\pi nx}{b}\right)=\lim\limits_{m\to\infty}\cos\left(\frac{\pi x}{b}(1+2^m)\right)\prod_{n=1}^m \cos\left(\frac{\pi x}{b}(2^{n-1})\right)$$ and $m$ is an integer.
