An identity about Gamma and Riemann function

[zetafunction]

we know that $$\Gamma (s)= \int_{0}^{\infty}dxe^{-x}x^{s-1}$$

however every factor of the Riemann Zeta can be obtained also from a Mellin transform

$$\int_{0}^{\infty}dxf(x)x^{s-1} =(1-p^{-s})^{-1}$$

where f(x) is the distribution

$$\sum_{n=0}^{\infty}x \delta (x-p^{-n})$$

is there any connection between Gamma and Riemann Zeta function i mean ,appart from appearing on the functional equation, also the Gamma function satisfy a relfection formula relating 's' and '1-s'

 

[fresh_42]

Here's a paper about it:
http://math.uchicago.edu/~may/REU2014/REUPapers/Duncan.pdf