Simon Plouffe found experimentally a series for $\pi$ that can be written as


$$\frac{\pi}{24} = \sum\_{n=1}^\infty \frac{1}{n} \left( \frac{3}{e^{\pi n}-1} -\frac{4}{e^{2\pi n}-1} +\frac{1}{e^{4\pi n}-1}\right) $$


A related series for $\log(2)$, easily found with lindep in PARI, is
$$\frac{\log(2)}{8} = \sum\_{n=1}^\infty \frac{1}{n} \left( \frac{2}{e^{\pi n}-1} -\frac{3}{e^{2\pi n}-1} +\frac{1}{e^{4\pi n}-1}\right) $$


Linas Vepstas proves several formulas similar to these in his paper [On Plouffe's Ramanujan Identities](https://arxiv.org/pdf/math/0609775.pdf), for constants of the form $\zeta(4m-1)$ , $\zeta(4m+1)$ and related powers of $\pi$.


However, regarding $\pi$ and $\log(2)$, the closest expression found in the reference is


$$S\_1(2\pi)+T\_1(2\pi) = \frac{\pi}{6} - \frac{3}{4}\log(2)$$


where
$$S\_1(x)=\sum\_{n=1}^\infty \frac{1}{n(e^{xn}-1)}$$
and
$$T\_1(x)=\sum\_{n=1}^\infty \frac{1}{n(e^{xn}+1)}$$


Therefore, there seems to be another similar equation that would allow for proving the above two series for $\pi$ and $\log(2)$, by eliminating for one constant or the other.



> 
> What is this second equation involving $\pi$ and $\log(2)$ ?
> 
> 
> 



Proving a series for $\pi$ by Plouffe
It is immediate to show that
$S\_1(x)=-x/24-\log(\eta(ix/(2\pi)))$
and $T\_1(x)=S\_1(x)-2S\_1(2x)$, so all these formulas are immediate
consequences of the properties of the Dedekind $\eta$ function.


