关于平均的Chowla猜想的动力学推广
Two averaged dynamical generalizations of Chowla's conjecture
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- School of Mathematics and Statistics, Yunnan University（云南大学数学与统计学院）
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中文总结 AI 辅助
该研究针对平均意义下的Chowla猜想给出动力学推广，借鉴Qi与Zheng在不可约二元三次型上的相关方法，证明了平均意义下的动力学Chowla猜想，还得到了素数序列上的类似结论。
中文摘要 AI 辅助
设k≥1为整数，λ为刘维尔函数。1965年Chowla提出猜想：对任意互不相同的自然数h₁,…,h_k，λ(n+h₁),…,λ(n+h_k)的值是渐近不相关的。本文受Bergelson与Richter关于素数定理的动力学推广的近期工作启发，将证明平均意义下Chowla猜想的动力学推广。证明中，我们采用Qi与Zheng建立的、适用于不可约二元三次型上的Bergelson-Richter定理变体的方法。此外，我们还将运用该方法证明素数序列上平均意义下的动力学Chowla猜想的类似结论。
英文摘要
Let $k\ge1$ be an integer and let $λ$ be the Liouville function. In 1965, Chowla gave a conjecture that the values of $λ(n+h_1),\dots, λ(n+h_k)$ are asymptotically unrelated for any distinct natural numbers $h_1, \dots, h_k$. In this article, motivated by the recent work of Bergelson and Richter on the dynamical generalizations of the prime number theorem, we will show a dynamical generalization of Chowla's conjecture on average. In the proof, we follow an approach of Qi and Zheng who established a variant of Bergelson and Richter's theorem over irreducible binary cubic forms. Moreover, we will use this approach to show an analogue of the dynamical Chowla's conjecture along the primes on average. In 2016, Tao proved that the two-point logarithmic Chowla's conjecture holds. Recently, Charamaras and Richter generalized Tao's theorem to bounded arithmetic functions and proposed a conjecture that generalizes Chowla's conjecture to bounded multi-variable arithmetic functions. Building on their work, we prove a dynamical generalization of Tao's theorem and a variant for the composition of the sum-of-digits function with the prime Omega function.
