Who's up to the challenge? 47

$\large \int_0^{2016} H_x^{(2)} \, dx$

Let ${ H }_{ n }^{ (2) }$ be the generalized harmonic number of order $n$ of 2, $\displaystyle H_n ^{(2)} = \sum_{k=1}^n \dfrac 1{k^2}$ .

If the integral above is equal to $a\zeta(b) - H_c$ , where $a,b$ and $c$ are positive integers, find $a+b+c$ .

Notations :

• $\zeta(\cdot)$ denotes the Riemann zeta function .

• $H_n$ denotes the $n^\text{th}$ harmonic number , $H_n = 1 + \dfrac12 + \dfrac13 + \cdots + \dfrac1n$ .