I have a sum of the form $$Q = \sum_{x}f(x).$$ Here, the $x$ entering the sum are the countably infinite set of solutions to $$ \tan(C x) = x.$$ The $x$ are not integers, but they do satisfy $$ \frac{\pi}{2C}(2n+1)\leq x_n \leq \frac{\pi}{2C}(2n+3),$$ where the $n$ are integers.
I am interested in the case of large $C$. In the limit of large $C$, successive solutions to $\tan(C x)=x$ become closer and closer together, separated by a gap $x_{n+1}-x_n = \pi/C$, so we can imagine writing a kind of Reimann sum for $Q$, although I am unclear how to do this.
Is it possible to write
$$Q \approx \int_{-\infty}^\infty f(x) w(x) dx ?$$ If so, how do I obtain the weight function $w(x)$?