Divisor Generating Function?

It is known that \[\large \displaystyle \sum _{n=1}^{\infty }\dfrac{\sigma_3(n)}{n^2}e^{-2\pi n}=\dfrac{G}{a} - \dfrac{b\pi^c}{d}\]

where \(G\) is denotes the Catalan's constant

\[\displaystyle G=\sum _{n=0}^{ \infty }\frac{\left(-1\right)^n}{\left(2n+1\right)^2},\]

\(\sigma_3(n)\) is defined to be the divisor function

\[\displaystyle σ_3(n)=\sum_{d\mid n}d^3,\]

And \(a,b,c\) and \(d\) are positive integers, with \(b,d\) coprime. Find \(a+b+c+d\).


Inspiration.

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