关于一类算术函数的$\boldsymbol{\text{Besicovitch}}$$\boldsymbol{\text{几乎周期性}}
On $\mathcal{B}^4$-almost periodicity for a class of arithmetical functions
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中文总结 AI 辅助
本文针对Chandrasekharan和Narasimhan提出的一类算术函数的归一化误差项，通过推导截断Voronoi型公式，证明其属于Besicovitch空间B⁴，强化了此前的几乎周期性结论，还得出误差项存在极限概率分布并给出四阶平均矩显式公式。
中文摘要 AI 辅助
本文针对Chandrasekharan和Narasimhan提出的一大类算术函数的适当归一化误差项，建立了Besicovitch意义下的$\boldsymbol{\text{几乎周期性}}。该结果显著强化了此前文献中针对相关误差项的$\boldsymbol{\text{几乎周期性结论}}。通过推导截断的Voronoi型公式，我们证明该归一化误差项属于Besicovitch空间$B^4$。作为推论，我们得出该误差项存在极限概率分布，并给出了其四阶平均矩的显式公式。
英文摘要
In this paper, we establish the $\mathcal{B}^4$-almost periodicity in the sense of Besicovitch for a suitably normalized error term associated with a broad class of arithmetical functions introduced by Chandrasekharan and Narasimhan. This result significantly strengthens $\mathcal{B}^2$-almost periodicity previously investigated in the literature for related error terms. By deriving truncated Voronoï-type formulas, we demonstrate that the normalized error term lies in the Besicovitch space $B^4$. As a consequence, we deduce that the error term admits a limit probability distribution and establish an explicit formula for its mean fourth power moment.
