Some pattern in the gap of the extrema/root of a product using Stirling approximation

Question

Playing with Desmos calculator and Stirling approximation I define :

$$x>0,f(x)=\prod_{n=1}^{M}\left(2-\frac{\Gamma(\frac{x}{n}+1)}{\sqrt{2\pi×\frac{x}{n}}\left(\frac{x}{e×n}\right)^{\frac{x}{n}}}\right)$$

If you plot it properly over $(1/2,5/4)$ say $M=30$ some patterns appears :

The extrema and roots are very regular in their successive gap and it's oscillating .

For $M=30$ the gap is around $3/5$

Question :

Can we show the existence of a constant or $constant\simeq root_{i+1}-root_{i}$ or the gap diverges like the harmonic sum ?

Answer 1

$x$ is a root of $f$ iff it makes one of the factors zero; that is, iff for some $n$ we have $g(x/n)=2$ where $g(t)=\frac{\Gamma(1+t)}{\sqrt{2\pi t}e^{-t}t^t}$.

It happens that the function $g$ is strictly decreasing and equals 2 at just one place, near $t=0.5728$. Call this $\tau$. So $g(x/n)=2$ when $x=n\tau$ for positive integer $n\leq M$. So yes, the roots are very regular indeed :-).