# A Floor Over A Floor!

Calculus Level 5

$\large \displaystyle \sum_{n=1}^{\infty} \frac{\lfloor \sqrt{n} \rfloor }{\lfloor \sqrt[4]{n} \rfloor ^7 }$

The above sum can be expressed as :

$\displaystyle a\zeta (\alpha) + b\zeta (\beta) + \frac{c}{d} \zeta (\gamma) + e \zeta (\delta) +\frac{f}{g} \zeta (\epsilon)$

Where $$a,b,c,d,e,f,g,\alpha,\beta,\gamma,\delta,$$ and $$\epsilon$$ are positive integers with $$c$$ and $$d$$ being coprime, $$f$$ and $$g$$ are also coprime.

Find : $$a+b+c+d+e+f+g+\alpha +\beta +\gamma +\delta +\epsilon$$.

Details and Assumptions:

$$\bullet \lfloor x \rfloor$$ Denotes the Floor Function.

$$\bullet \zeta ( x)$$ is the Riemann Zeta Function.

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