Almost sure bounds for weighted sums of Rademacher random multiplicative functions

We prove that when $f$ is a Rademacher random multiplicative function for any $\epsilon>0$, then $\sum_{n \leqslant x}\frac{f(n)}{\sqrt{n}} \ll (\log\log(x))^{3/4+\epsilon}$ for almost all $f$. We also show that there exist arbitrarily large values of $x$ such that $\sum_{n \leqslant x}\frac{f(n)}{\sqrt{n}} \gg (\log\log(x))^{-1/2}$. This is different to what is found in the Steinhaus case, this time with the size of the Rademacher Euler product making the multiplicative chaos contribution the dominant one. We also find a sharper upper bound when we restrict to integers with a prime factor greater than $\sqrt{x}$, proving that $\sum_{\substack{n \leqslant x \\ P(n) > \sqrt{x}}}\frac{f(n)}{\sqrt{n}} \ll (\log\log(x))^{1/4+\epsilon}$.