Closed formula for some cotangent series I managed to compute some cotangent series, such as $~\displaystyle\sum\limits_{{\substack{i=1\\10~\nmid~i}}}^{\infty}\frac{\cot\left(\dfrac{9\pi}{10}\cdot i\right)}{i}=-\frac{6\pi}{5},~$ and $\displaystyle\sum\limits_{{\substack{i=1\\11~\nmid~i}}}^{\infty}\frac{\cot\left(\dfrac{4\pi}{11}\cdot i\right)}{i}=\frac{3\pi}{11}.~$ I am interested in the general case 

$$S_{k,~n}~=~\sum\limits_{{\substack{i=1\\n~\nmid~i}}}^{\infty}\frac{\cot\left(\dfrac{k\pi}{n}\cdot i\right)}{i}$$ 

where $k,~n$ are positive integers with $k<n$ and $\gcd(k,n)=1.~$ References about a closed form (if there is one) or some nice properties will be appreciated. 
 A: This answer is only a hint, the formula can still be simplified.
It’s given $ \enspace\displaystyle  S_{k,n}|_{gcd(k,n)=1}=\sum\limits_{{\substack{j=1\\n~\nmid~j}}}^\infty\frac{1}{j}\cot\left(\frac{k\pi}{n}j\right) \enspace $ which means 
$\enspace\displaystyle  S_{k,n}=\sum\limits_{m=1}^\infty\frac{a_m}{m}\enspace $ with $\enspace\displaystyle a_{m+ln}=a_m:=\cot\left(\frac{k\pi}{n}m\right)\enspace$ and $\enspace a_{ln}=a_n:=0\enspace $ , $\enspace  l\in\mathbb{N}_0$.
Using $ \enspace\displaystyle E_1(x):=\sum\limits_{k=1}^\infty \frac{1}{k}e^{i2\pi kx}=i\pi\left(\frac{1}{2}-x\right)-\ln(2\sin(\pi x)) \enspace $ for $\enspace 0<x<1\enspace $ and https://www.fernuni-hagen.de/analysis/download/bachelorarbeit_aschauer.pdf, page 4 (2.1), 
one gets $\enspace\displaystyle  b_m |_{n\nmid m}=\frac{1}{n}\sum\limits_{v=1}^{n-1}a_v e^{-i2\pi \frac{m}{n}v}\enspace$ and additional  $\enspace\displaystyle  b_n=\frac{1}{n}\sum\limits_{v=1}^{n-1}a_v =0\enspace$  and therefore $\enspace\displaystyle \sum\limits_{m=1}^\infty\frac{a_m}{m}=\sum\limits_{m=1}^{n-1}b_m E_1\left(\frac{m}{n}\right)\enspace $. 
Result:
$$S_{k,n}=\frac{1}{n}\sum\limits_{m=1}^{n-1} \left(i\pi\left(\frac{1}{2}-\frac{m}{n}\right)-\ln\left(2\sin\frac{m\pi}{n}\right)\right)\sum\limits_{v=1}^{n-1}\cot\left(\frac{k\pi}{n}v\right) e^{-i2\pi \frac{m}{n}v}$$

Note: 
A simple numerical test with WolframAlpha gives $11 S_{4,11}\approx 9.42478$ which means $3\pi$ ; input: 
sum (i pi (0.5-m/11)-ln(2 sin((m pi)/11)))*(sum (cot(4 pi v/11) e^(-i 2 pi m v/11))  for v=1 to 10)  for m=1 to 10
