Can it be shown that there is, or is not, a unique, and logarithmically convex function that generalises the Primorial to non-natural numbers in the way the Gamma function generalises the factorial?
User Drax gives a very nice generalisation here which may or may not be it but assessing log convexity of that looks tricky.
The reason the question's interesting to me is that I'm curious to learn more about the convergence of the imaginary part of $n\cdot\exp\left(\frac{2\pi\cdot i}{\log_{n}(p_n\#)}\right)$ to $2\pi i$ as $n\to\infty$ and having a smooth function seems the best way to gain more insight. Picking out the most natural unique such example seems a logical start.
