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Calculus Level 5

Let \( \displaystyle { H }_{ n }^{ (3) }=\sum _{ k=1 }^{ n }{ \frac { 1 }{ { k }^{ 3 } } } \). If \[\sum _{ n=1 }^{ \infty }{ \frac { { H }_{ n }^{ (3) } }{ n^{ 2 } } } \]

is in the form \[\dfrac { a\zeta (b) }{ c } -\dfrac { \pi ^{ d }\zeta (f) }{ g } \]

for positive integers \(a,b,c,d,f\) and \(g\) where \(a\) and \(c\) are coprime, find \(a+b+c+d+f+g\).

Notation: \(\zeta(\cdot) \) denotes the Riemann zeta function.

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