Another Series $\sum\limits_{k=2}^\infty \frac{\log(k)}{k}\sin(2k \mu \pi)$ I ran across an interesting series in a paper written by J.W.L. Glaisher. Glaisher mentions that it is a known formula but does not indicate how it can be derived.
I think it is difficult.
$$\sum_{k=2}^\infty \frac{\log(k)}{k}\sin(2k \mu \pi) = \pi \left(\log(\Gamma(\mu)) +\frac{1}{2}\log \sin(\pi \mu)-(1-\mu)\log(\pi)- \left(\frac{1}{2}-\mu\right)(\gamma+\log 2)\right)$$
Can someone suggest a method of attack?
$\gamma$ is the Euler-Mascheroni Constant.
Thank You!

That is the well known Fourier-expansion of $\log\Gamma(x)$ doing by Kummer in 1847. The article of Glaisher is from 1893. See
https://de.wikipedia.org/wiki/Gammafunktion
