In the answer here, I showed that $$f(m,n)=\int_0^{\pi/2}\ln \lvert\sin(mx)\rvert \cdot \ln \lvert\sin(nx)\rvert \, dx = \frac{\pi^3}{24} \frac{\gcd^2(m,n)}{mn}+\frac{\pi\ln^2(2)}{2}$$ This identity, sadly, only evaluates the integral for positive integer value of $m,n$. I would like to know if anyone can figure out how to evaluate it for other values of $m,n$ - perhaps rational numbers like $1/3$, or, even better, irrational numbers like $\pi, e, \sqrt 2$. I am really just looking for special values.
Also, I think this converges for nonzero $m,n$, but because of all of the poles of the integrand potentially contained within the interval of integration, no software of mine will plot this in three dimensions (or even very well in two dimensions, with one of the variables held constant) as a function of $m,n$.
Can anyone produce any special values, good graphs, or interesting properties of $f$?
