<s>[INST] Does $\sum\limits_{k \geq 0} \frac{1}{(4k+1)(4k+2)} = \frac{\log(2)}{4} + \frac{\pi}{8}$ hold? Context: I'm interested in trying to find as many explicit formulae as possible for:
$$ S_n(s) = \sum_{k \geq 0} \frac{1}{\prod_{i \in s} (nk+i)} $$
for all $ n \geq 2 $ and for all $ s $ subset of $ \{ 1, \dots, n \} $ such that $ \mbox{Card}(s) = 2 $.
So far, I can establish
$$ S_2(\{ 1, 2 \}) = \log(2) $$
$$ S_3(\{ 1, 2 \}) = \frac{\pi \sqrt{3}}{9},
S_3(\{ 2, 3 \}) = \frac{\log\left(3\right)}{2}-\frac{\pi\sqrt{3}}{18},
S_3(\{ 1, 3 \}) = \frac{\log\left(3\right)}{4}+\frac{\pi\sqrt{3}}{36} $$
$$ S_4(\{ 1, 3 \}) = \frac{\pi}{8}, S_4(\{ 2, 4 \}) = \frac{\log(2)}{4} $$
But I just guess that
$$ S_4(\{ 1, 2 \}) = \frac{\log(2)}{4} + \frac{\pi}{8} $$
Is there a generic method to establish all formulae?
 [/INST] $$\begin{aligned} S&= \sum_{k\geq 0}\frac{1}{(4k+1)(4k+2)} \\& =\sum_{k\geq 0}\left(\frac{1}{4k+1}-\frac{1}{4k+2}\right)\\& =\frac{1}{4}\sum_{k\geq 0} \left(\frac{1}{k+\frac{1}{4}}-\frac{1}{k+\frac{1}{2}}\right)=\frac{1}{4}\left(\psi^0\left(\frac{1}{2}\right)-\psi^0\left(\frac{1}{4}\right)\right)\\& =\frac{1}{4}\left(-\gamma -\ln(4)+\gamma+\frac{\pi}{2}+\ln(8)\right)=\frac{\ln2}{4} +\frac{\pi}{8}\end{aligned} $$
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