Relations between the random variable $w_x$ and the Dirichlet divisor problem

We have developed a heuristic showing that in the Dirichlet divisor problem for the almost all $n \in \mathbb{N}^{+}$: $$R(n) \leq O(\psi(n)n^{\frac{1}{4}})$$ where $$R(n) = \Big\lvert \sum_{x=1}^{n}\Big\lfloor\frac{n}{x}\Big\rfloor - n\log{n} - (2\gamma-1)n \Big\rvert$$ and $\psi(n)$ - any positive function that increases unboundedly as $n \to \infty$. The result is achieved under the hypothesis: $$\Big \{\frac{n}{x} \Big \} \sim w_x$$ where $w_x$ is uniformly distributed over $[0,1)$ random variable with a values set $\{0, \frac {1} {x}, \ldots, \frac{x-1}{x} \}$ and the value accepting probability $p = \frac{1}{x}$. The paper concludes with a numerical argument in support of the hypothesis being true. It is shown that the expectation: $$\mu_{1} \Big[\sum_{x=1}^{n}\Big(\frac{n}{x} - \frac{x-1}{2x}\Big) \Big]= (2n+1)H_{\lfloor\sqrt{n}\rfloor} - \lfloor\sqrt{n}\rfloor^{2} - \lfloor\sqrt{n}\rfloor + C$$ has deviation from $D(n)$ is less than $R(n)$ in absolute value for all $n < 10^{5}$.