素数定理的动力学推广以及加法与乘法半群作用的不交性
Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions
AI总结：
本研究建立两个蕴含多项乘性数论经典结果的遍历定理，将Sarnak的Möbius不交性猜想重构于新动力学框架，提出聚焦加乘半群作用不交性的猜想扩展并证明其若干特例。
摘要：
我们建立了两个遍历定理，其推论包含众多来自乘性数论的经典结果，包括素数定理、Pillai-Selberg定理、Erdős-Delange定理、Wirsing均值定理以及Halász均值定理的特殊情形。借助遍历结果背后的思路，我们将Sarnak的Möbius不交性猜想重构于一个新的动力学框架中。这自然引出了Sarnak猜想的一个扩展，该扩展聚焦于加法与乘法半群作用的不交性。我们通过证明若干特殊情形来佐证这一扩展的合理性。
英文摘要：
We establish two ergodic theorems which have among their corollaries numerous classical results from multiplicative number theory, including the Prime Number Theorem, a theorem of Pillai-Selberg, a theorem of Erdős-Delange, the mean value theorem of Wirsing, and special cases of the mean value theorem of Halász. By building on the ideas behind our ergodic results, we recast Sarnak's Möbius disjointness conjecture in a new dynamical framework. This naturally leads to an extension of Sarnak's conjecture which focuses on the disjointness of additive and multiplicative semigroup actions. We substantiate this extension by providing proofs of several special cases.
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