Geometric mean of prime gaps? The arithmetic mean of prime gaps around $x$ is $\ln(x)$.
What is the geometric mean of prime gaps around $x$ ?
Does that strongly depend on the conjectures about the smallest and largest gap such as Cramer's conjecture or the twin prime conjecture ?

I thought Hardy-Littlewood might come into it.
Here is some numerical data following Erics great answer:

x-axis: N 
y-axis: Geometric mean of the first 10000 prime gaps following $10^N$ divided by $\ln 10^N$.
$e^{-\gamma} \approx 0.56146$.
