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!
 A: It suffices to do these integrals:
$$
\begin{align}
\int_0^1 \log(\Gamma(s))\;ds &= \frac{\log(2\pi)}{2}
\tag{1a}\\
\int_0^1 \log(\Gamma(s))\;\cos(2k \pi s)\;ds &= \frac{1}{4k},\qquad k \ge 1
\tag{1b}\\
\int_0^1 \log(\Gamma(s))\;\sin(2k \pi s)\;ds &= \frac{\gamma+\log(2k\pi)}{2k\pi},\qquad k \ge 1
\tag{1c}
\\
\int_0^1 \frac{\log(\sin(\pi s))}{2}\;ds &= \frac{-\log 2}{2}
\tag{2a}
\\
\int_0^1 \frac{\log(\sin(\pi s))}{2}\;\cos(2k \pi s)\;ds &= \frac{-1}{4k},\qquad k \ge 1
\tag{2b}
\\
\int_0^1 \frac{\log(\sin(\pi s))}{2}\;\sin(2k \pi s)\;ds &= 0,\qquad k \ge 1
\tag{2c}
\\
\int_0^1 1 \;ds &= 1
\tag{3a}
\\
\int_0^1 1 \cdot \cos(2k \pi s)\;ds &= 0,\qquad k \ge 1
\tag{3b}
\\
\int_0^1 1 \cdot \sin(2k \pi s)\;ds &= 0,\qquad k \ge 1
\tag{3c}
\\
\int_0^1 s \;ds &= \frac{1}{2}
\tag{4a}
\\
\int_0^1 s \cdot \cos(2k \pi s)\;ds &= 0,\qquad k \ge 1
\tag{4b}
\\
\int_0^1 s \cdot \sin(2k \pi s)\;ds &= \frac{-1}{2k\pi},\qquad k \ge 1
\tag{4c}
\end{align}
$$
Then for $f(s) = \pi \left(\log(\Gamma(s)) +\frac{1}{2}\log \sin(\pi s)-(1-s)\log(\pi)- \left(\frac{1}{2}-s\right)(\gamma+\log 2)\right)$, we get
$$
\begin{align}
\int_0^1 f(s)\;ds &= 0
\\
2\int_0^1f(s) \cos(2k\pi s)\;\;ds &= 0,\qquad k \ge 1
\\
2\int_0^1f(s) \sin(2k\pi s)\;\;ds &= \frac{\log k}{k},\qquad k \ge 1
\end{align}
$$
and the formula follows as a Fourier series:
$$
f(s) = \sum_{k=1}^\infty \frac{\log k}{k}\;\sin(2 k\pi s),\qquad 0 < s < 1.
$$  
reference 
Gradshteyn & Ryzhik, Table of Integrals Series and Products
(1a) 6.441.2
(1b) 6.443.3
(1c) 6.443.1
(2a) 4.384.3
(2b) 4.384.3
(2c) 4.384.1  
A: 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
