带全平方核的函数与不变平均函数之间的渐近不相关性
Asymptotic uncorrelations between functions with squarefull kernel and functions of invariant average
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中文总结 AI 辅助
本文研究带全平方核的$s$-函数与唯一遍历系统中沿素因子计数函数的轨道的渐近相关性，证明二者渐近不相关，改进了素数定理并得到相关变体。
中文摘要 AI 辅助
1986年，Ivić和Tenenbaum引入了带全平方核的算术函数，也称为$s$-函数。后来，Erdős和Ivić给出了$s$-函数移位卷积和的渐近估计。最近，Bergelson和Richter研究了唯一遍历拓扑动力系统中沿素因子计数函数（prime Omega function）的轨道，并建立了素数定理（PNT）的新动力推广，这些轨道可视为乘法下的不变平均函数。本文证明，$s$-函数及其移位卷积，与唯一遍历系统中沿素因子计数函数的轨道渐近不相关。由此，通过$s$-函数的局部分布，得到了素数定理（PNT）的一个改进结果，同时也建立了这些结果的若干变体。
英文摘要
In 1986, Ivić and Tenenbaum introduced arithmetic functions with squarefull kernel, which are also called $s$-functions. Later, Erdős and Ivić gave an asymptotic estimate on the shifted convolution sums of $s$-functions. Recently, Bergelson and Richter studied the orbits along the prime Omega function in a uniquely ergodic topological dynamical system and established a new dynamical generalization of the prime number theorem (PNT). These orbits can be viewed as functions of invariant average under multiplications. In this paper, we show that both $s$-functions and their shifted convolutions are asymptotically uncorrelated to the orbits along the prime Omega function in a uniquely ergodic system. As a consequence, we obtain a refinement of the PNT via the local distribution of $s$-functions. Furthermore, several variants of these results are established as well.
