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

TL;DR

This paper proposes a heuristic linking the distribution of fractional parts of n/x to a random variable, providing bounds on the Dirichlet divisor problem's error term for most n, supported by numerical evidence.

Contribution

It introduces a novel heuristic connecting fractional parts to a uniformly distributed random variable, offering a new approach to estimate the divisor problem's error term.

Findings

Heuristic suggests fractional parts are modeled by a uniform distribution.

Numerical evidence supports the hypothesis for n < 10^5.

Provides bounds on the error term R(n) in the divisor problem.

Abstract

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 $n < 10^{5}$.