Relation between Gamma Function and Zeta Function

We know that $\Gamma(x) = \int_0^\infty e^{-t}t^{x-1} dt$

Let $t = ru \implies dt = r du$ Which means

$\Gamma(x) = \int_0^\infty e^{-ru} {(ru)}^{x-1} r du$

$\Gamma (x) = \int_0^\infty r^x u^{x-1} e^{-ru} du$

$\frac{1}{r^x} \Gamma (x) = \int_0^\infty u^{x-1} e^{-ru} du$

Taking sum on both the sides we get

$\sum_{r=0}^\infty \frac{1}{r^x} \Gamma (x) = \sum_{r=0}^{\infty}\int_0^\infty r^x u^{x-1} e^{-ru} du$

$\Gamma(x) \sum_{r = 0}^{\infty} \frac{1}{r^x} = \int_0^{\infty}u^{x-1} \sum_{r=0}^{\infty}e^{-ru} du$

$\Gamma(x) \zeta(x) = \int_{0}^{\infty}u^{x-1}\frac{e^{-u}}{1-e^{-u}} du$

$\Gamma(x) \zeta(x) = \int_{0}^{\infty} \frac{u^{x-1}}{e^u-1}du$

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