<s>[INST] General form for $\sum_{n=1}^{\infty} (-1)^n \left(m n \, \text{arccoth} \, (m n) - 1\right)$ I'm wondering if there is a general form for the following sum:

$$\sum_{n=1}^{\infty} (-1)^n \left(m n \, \text{arccoth} \, (m n) - 1\right)$$ for $m \in \mathbb{N}$

I have obtained the following closed-forms for these special cases:

Where $G$ is Catalan's constant and $\text{Cl}_2$ is the Clausen function of order 2.
$$\sum_{n=1}^{\infty}(-1)^n \left(2n \, \text{arccoth} \, (2n) - 1\right)  = \frac{1}{2} - \frac{2G}{\pi}$$
$$\sum_{n=1}^{\infty} (-1)^{n}\left( 3n \, \text{arccoth} \, (3n)-1\right)  = \frac{1}{2} - \frac{5}{2\pi} \, \text{Cl}_2 \, \left(\frac{\pi}{3}\right) + \frac{1}{4} \ln (3)$$
$$\sum_{n=1}^{\infty} (-1)^n \left(4n \, \text{arccoth} \, (4n) - 1\right) = \frac{1}{2}+ \frac{G}{\pi} - \frac{4}{\pi} \, \text{Cl}_2 \, \left(\frac{\pi}{4}\right) - \frac{1}{4} \ln \left(3- 2\sqrt{2}\right)$$
etc.

Given that $G = \text{Cl}_2 \left(\frac{\pi}{2}\right)$, I am curious to know if the general sum is expressible in terms of the Clausen function. These above sums were determined by using the Mittag-Leffler expansion of $\csc (z)$, i.e $\csc(z) = \frac{1}{z} + 2z \sum_{n=1}^{\infty} (-1)^n \frac{1}{z^2 - \left(\pi n\right)^2}$ and substituting it into the integral $\int_{0}^{\pi/m} x \csc (x) \, dx$
If one uses the following other method, we can determine the odd and even terms of the sums

$$\sum_{n=1}^{\infty} \left( 4n \, \text{arccoth} \, (4n)-1\right) = \frac{1}{2} - \frac{G}{\pi}- \frac{1}{4} \ln (2)$$
$$\sum_{n=1}^{\infty} \left( (4n-2) \, \text{arccoth} \, (4n-2) - 1\right) = \frac{G}{\pi} - \frac{1}{4} \ln (2)$$
$$\sum_{n=1}^{\infty} \left(6n \, \text{arccoth} \, (6n) - 1\right) = \frac{1}{2} - \frac{3}{2\pi} \, \text{Cl}_2 \left( \frac{\pi}{3}\right)$$
$$\sum_{n=1}^{\infty} \left( (6n-3) \, \text{arccoth} \, (6n-3) - 1\right) = \frac{1}{\pi} \, \text{Cl}_2 \, \left(\frac{\pi}{3}\right) - \frac{1}{4} \ln(3)$$
$$\sum_{n=1}^{\infty} \left( 8n \, \text{arccoth} \, (8n) - 1\right) = \frac{1}{2} - \frac{2}{\pi} \, \text{Cl}_2 \, \left(\frac{\pi}{4}\right) - \frac{1}{4} \ln (2-\sqrt{2})$$
$$\sum_{n=1}^{\infty} \left( (8n-4) \, \text{arccoth} \, (8n-4) - 1\right) = \frac{2}{\pi} \, \text{Cl}_2 \, \left(\frac{\pi}{4}\right) - \frac{G}{\pi} - \frac{1}{4} \ln(2+\sqrt{2})$$

EDIT

I am now interested in
$$\sum_{n=1}^{\infty} \left(m n \, \text{arccoth} \, (m n) - 1\right)$$
for $|m|>1$ as asked here.


Here is how I originally proved it for the odd terms:
For the first odd term sum, begin with the known result $$2G = \int_{0}^{\frac{\pi}{2}} x \csc (x) \, dx$$
then integrate by parts to get:
$$2G = \int_{0}^{\frac{\pi}{2}} \ln \left(\cot (x) + \csc (x) \right) \, dx = \int_{0}^{\frac{\pi}{2}} \ln (\cos (x) + 1) \, dx - \int_{0}^{\frac{\pi}{2}} \ln (\sin (x)) \,dx$$
Then use the well-known result $\int_{0}^{\frac{\pi}{2}} \ln (\sin(x)) \,dx = - \frac{\pi}{2} \ln (2)$ and use the identity $\cos (x) + 1 = 2 \cos^2\left( \frac{x}{2}\right)$ and make the substitution $\frac{x}{2} = u$.
$$\implies 2G - \pi \ln (2) = 4 \int_{0}^{\frac{\pi}{4}} \ln (\cos (u)) \, du$$
Now use the Weierstrass product for $\cos (z)$, namely $\cos(z) = \prod_{n=1}^{\infty} \left(1-\frac{4z^2}{\pi^2 (2n-1)^2}\right)$ to obtain:
$$2G - \pi \ln (2) = 4 \sum_{n=1}^{\infty} \int_{0}^{\frac{\pi}{4}} \ln \left( 1-\frac{4u^2}{\pi^2 (2n-1)^2}\right) \, du$$
After integrating, obtain $\pi \sum_{n=1}^{\infty} \ln \left(1-\frac{1}{4(1-2n)^2}\right) = -\frac{\pi}{2} \ln (2)$ and the result quickly follows. The other odd sums are the same idea. The even sums just comes from combining the two results from the alternating sum and the odd term sum.
 [/INST] $\newcommand{\D}{\mathfrak{D}}\newcommand{\d}{\,\mathrm{d}}$Contour integration naturally leads us to the situation of Random Variable’s answer.
Fix a real $s>1$ and call this series $S$. Let $\log$ be the principal logarithm, so that $z\mapsto\log\frac{zs+1}{zs-1}$ is holomorphic on $\Bbb C\setminus[-s^{-1},s^{-1}]$. Note that this interval is strictly contained in the open unit disk since $s>1$.
For $0<\epsilon<s^{-1}$ let $\D_\epsilon$ be the contour which runs: $-s^{-1}+i\epsilon\to s^{-1}+i\epsilon$ in a straight line, round in a clockwise semicircle of radius $\epsilon$ to $s^{-1}-i\epsilon$, from there to $-s^{-1}-i\epsilon$ in a straight line and back in another clockwise semicircle to $-s^{-1}+i\epsilon$: a “dogbone contour”.
Define $f:\Bbb C\to\hat{\Bbb C}$ by: $$z\mapsto\frac{zs}{2}\log\frac{zs+1}{zs-1}-1$$
Since $f$ satisfies the growth condition $zf(z)\to0$ uniformly as $|z|\to\infty$ we have, using the usual large box contour with corners $2\pi(N+1/2)(\pm1\pm i)$ for large $N$: $$4\pi i S=\oint_{\D_\epsilon}\pi\csc(\pi z)f(z)\d z$$For any such $\epsilon$.
Extracting the “$-1$” out of $f$, $\oint_{\D_\epsilon}\pi\csc(\pi z)(-1)\d z=2\pi i$ always holds. For $-1<x<1$ let $x^{\pm}$ denote a limit as $z\to x$ from above and below the real axis. Then the branch jump can be, with simple algebra, found to be: $$\log\frac{x^+s+1}{x^+s-1}-\log\frac{x^-s+1}{x^-s-1}=-2\pi i$$Since we are free to take $\epsilon\to0^+$, this information gives - after dividing by $4\pi i$ - the equation: $$S=\frac{1}{2}-\frac{\pi s}{4}\int_{-s^{-1}}^{s^{-1}}x\csc\pi x\d x$$
The exact same method gives: $$S’=\frac{1}{2}-\frac{\pi s}{4}\int_{-s^{-1}}^{s^{-1}}x\cot\pi x\d x$$If $S’$ is the non-alternating series.
Then Random Variable’s answer(s) expresses these last integral via the Clausen function.
I find it prettier to rewrite these as: $$\begin{align}2\sum_{n\ge1}(\pi ns\cdot\operatorname{arcoth}(\pi ns)-1)&=1-\int_0^{s^{-1}}\phi\cot\phi\d\phi\\2\sum_{n\ge1}(-1)^n(\pi ns\cdot\operatorname{arcoth}(\pi ns)-1)&=1-\int_0^{s^{-1}}\phi\cdot\csc\phi\d\phi\end{align}$$
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