Various questions on MO concerning the "surprise" occurrence of the gamma function in the functional equation of the Riemann zeta function got me wondering whether the Gamma function alone suffice for an analytic formulation of the fundamental theorem of arithmetic.
Pondering this lead me to consider the product
whose convergence is apparently equivalent to the prime number theorem. (Fix $x\not=0,-1$, apply $\ln$ and linearize to get a series that eventually behaves like $1 - 1/2 - 1/3 -1/5 + 1/6 - 1/7 +\cdots$.) The limit appears to equal $e^x$.
Of course my motivation lay in using Mobius-type inclusion-exclusion to eliminate one by one all the poles.
My question: does this expression occur in the literature?