论文标题
随机变量$ w_x $与dirichlet除数问题之间的关系
Relations between the random variable $w_x$ and the Dirichlet divisor problem
论文作者
论文摘要
我们已经开发了一种启发式方法，表明在Dirichlet Divisor问题中，几乎所有$ n \ in \ Mathbb {n}^{+} $：$$ r（n）\ leq o（ψ（n）n^{\ frac {\ frac {1} {4} {4} {4}} {4}} {4}} {4}} {4}） \ sum_ {x = 1}^{n} \ big \ lfloor \ frac {n} {x} {x} \ big \ rfloor -n \ log {n} - （2γ-1）n \ big \ big \ rvert $ $ $ \ rvert $$和$ψ（n）$ - 任何积极的功能都会增加无符号的$ n \ n.结果是在假设下实现的：$$ \ big \ {\ frac {n} {x} \ big \} \ sim w_x $$其中$ w_x $均匀分布在$ [0,1）$随机变量上，值set $ \ \ \ frac {0，\ frac {1，\ frac {1，\ frac {1} {1} {1} \ frac {x-1} {x} \} $和值接受概率$ p = \ frac {1} {x} $。 本文以数值论点为支持该假设是真实的。结果表明，期望：$μ_{1} \ big [\ sum_ {x = 1}^{n} \ big（\ frac {n} {x} {x} {x} - \ frac {x -1} {2x} {2x} \ big] （2n + 1）h _ {\ lfloor \ sqrt {n} \ rfloor} - \ lfloor \ lfloor \ sqrt {n} \ rfloor^{2} - \ lfloor \ lfloor \ sqrt \ sqrt \ sqrt {n} \ rfloor + c $ n oby $ in $ in $ in $ in $ in n obly in nime in nime in y Insival in Oblial in Obraply（in）所有$ n <10^{5} $。
We have developed a heuristic showing that in the Dirichlet divisor problem for the almost all $n \in \mathbb{N}^{+}$: $$ R(n) \leq O(ψ(n)n^{\frac{1}{4}}) $$ where $$ R(n) = \Big\lvert \sum_{x=1}^{n}\Big\lfloor\frac{n}{x}\Big\rfloor - n\log{n} - (2γ-1)n \Big\rvert $$ and $ ψ(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: $$μ_{1} \Big[\sum_{x=1}^{n}\Big(\frac{n}{x} - \frac{x-1}{2x}\Big) \Big]= (2n+1)H_{\lfloor\sqrt{n}\rfloor} - \lfloor\sqrt{n}\rfloor^{2} - \lfloor\sqrt{n}\rfloor + C$$ has deviation from $D(n)$ is less than $R(n)$ in absolute value for all $n < 10^{5}$.
