How to express the Riemann hypothesis in terms of the Gamma function? 
*

*The Riemann hypothesis (RH) states that all non-trivial zeros of the zeta function have real part $\frac{1}{2}$.

*The zeta function is intimately connected with the Gamma function via the functional equation.
The second fact suggests that there is an equivalent form of RH which is expressed solely in terms of the Gamma function.
Question: What is the most natural form to translate RH as directly as possible (without mentioning the zeta function) into a hypothesis on the behaviour of the Gamma function?
 A: Since 
$$\zeta(z)=\frac{\Gamma (1-z) \left(2^{-z} \left(\psi \left(z-1,1\right)+\psi \left(z-1,\frac{1}{2}\right)\right)-\psi(z-1,1)\right)}{\ln(2)}$$
where $\psi(x,z)$ is the generalized polygamma following Espinosa's generalization, whatever we say about Zeta function we can also say about the right hand part of this identity. It consists only of Gamma function, its (fractional) derivatives and integrals.
A: The Wikipedia article gives a Mellin transform
$$\Gamma(s)\zeta(s) =\int_0^\infty\frac{x^{s-1}}{e^x-1} dx.$$
The Dirichlet series over the Möbius function gives the reciprocal 
$$ \frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^s} .$$
Thus we may write
$$\Gamma(s) = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^s} \int_0^\infty\frac{x^{s-1}}{e^x-1} dx .$$
This holds true for every complex number s with real part greater than $1$. Now let's try to enlarge the domain of validity of this representation. Riemann showed (see the book of H. M. Edwards, Riemann Zeta Function, for the details) that modifying the contour gives a formula valid for all complex s.
$$ 2\sin(\pi s)\Gamma(s)\zeta(s) = i \oint_C \frac{(-x)^{s-1}}{e^x-1}dx .$$
This leads to 
$$ \sin(\pi s) \Gamma(s) = \frac{i}{2} \sum_{n=1}^{\infty} \frac{\mu(n)}{n^s} 
\oint_C \frac{(-x)^{s-1}}{e^x-1} dx . \qquad (*)  $$
However, this formula is again only valid for s with real part greater than $1$ because of the use of the Dirichlet series. Wikipedia remarks: 

"The Riemann hypothesis is equivalent to the claim that [this
  representation of the reciprocal of the zeta function] is valid when
  the real part of $s$ is greater than $\frac{1}{2}$." 

Thus a possible answer to my question is:

The representation $\,*\,$ is valid for all $s$ with real part greater than $\frac{1}{2}$ if and only if the RH holds.

Perhaps someone can elaborate further to give this relation a more geometric meaning? Where are the non-trivial zeros of the zeta function to be spotted in this setup? 
A: I don't think it's possible.
There is a paper that expresses very good approximations of zeta using truncated euler products and gamma functions.
The author in this paper, uses finite euler products, gamma functions and forces functional equation on the approximation, subtracts the principal component part from the approximation, and some other tricks. The approximation is very efficient to compute the zeros of Zeta. However, the principal part on the imaginary line is an infinite sum of of reciprocal functions with coefficients involving gamma values, and a finite number of primes. The regularization is based on the observation that the euler product should have a natural boundary at sigma=0, but the analytical continuation of zeta has none on the imaginary axis. Hence the principal part of the euler product must vanish somehow, and the author simply removed it cleverly.
However, I don't think you can express the principal part as finite sum of gamma functions, but as a contour in integral yes of course.
But why even bother, when you already have a hankel contour integral and mellin integral representation for the zeta function in terms of the exponential.
