Let $\mu(n)$ the Möbius function, see it you need this Wikipedia. And here $n!$ is the factorial.
After I've known a similar question from MathOverflow, I would like to know if it is possible to deduce something about the question that I've created.
Question. Is $$\sum\limits_{n=1}^\infty \mu(n)\frac{x^n}{\sqrt{n!}}\tag{1}$$ positive, when $0<x<1$? Many thanks.
If my question was in the literature answer this question as a reference request, and I try to search and read those calculations.
My belief from my calculations using Wolfram Alpha online calculator (it is as a toy model of previous function in $(1)$) is that our function has a root in the unit interval, see it with this code
plot sum mu(n)/sqrt(n!) x^n, from n=1 to 1000, for 0<x<1