Let for integers $n\geq 1$ the Möbius function $\mu(n)$, and $p_n$ the $nth$ prime number.
I've read this Wikipedia's article about the Andrica's conjecture.
After I did few experiments with codes like this
sum mu(n)((Prime(n+1)^1)-(Prime(n))^1), from n=1 to 20000
sum mu(n)((Prime(n+1)^(1/2))-(Prime(n))^(1/2)), from n=1 to 1000
sum mu(n)((Prime(n+1)^(1/2))-(Prime(n))^(1/2)), from n=1 to 20000
with Wolfram Alpha online calculator, I am wondering that maybe there exists a positive constant $C>0$ such that $$ \left| \sum_{n\leq x}\mu(n)\left(\sqrt{p_{n+1}}-\sqrt{p_{n}}\right)\right|<C. $$
I don't know if this question or questions like this (I am saying using the function $x^{\frac{1}{k}}=\sqrt[k]{x}$ for a fixed integer $k>2$, instead of $\sqrt{x}$ in the expressions $\sqrt{\text{prime}}$ was/were in the literature or are interestings).
Question. Can you prove or refute that there exists a positive constant $C>0$ such that $$ \left| \sum_{n\leq x}\mu(n)\left(\sqrt{p_{n+1}}-\sqrt{p_{n}}\right)\right|<C$$ holds for all $x\geq 1$. Many thanks.
Additionally if you want (thus it is optional) please add remarks about if previous Question is interesting: Feel free to criticize it. Is it possible dilucidate the question? Was right the combination that I did of these arithmetic functions by means of partial sums with the purpose to set an interesting conjecture? If questions like as previous in Question or the generalization that I've evoked are in the literature feel free to refer those.