奇数主序列的 p-复合数作为 $\pi(x)$ 的构建块
p-composites of the main sequence of odd numbers as building blocks for $π(x)$
AI总结：
本文提出基于循环群的Z-函数算法，以O(x^{1/2})时间复杂度计算素数计数函数π(x)，并系统介绍升序分解、奇数主序列、T-级数等多种构建方法。
摘要：
素数计数函数 $\pi(x)$ 返回小于或等于给定数的素数个数，是数论中一个引人关注的主题。为了超越其同类算法，有人提出了一种基于与 $Z/nZ$ 同构的循环群的算法，即所谓的 $Z$-函数。该方法建议的时间复杂度为 $\mathcal{O}(x^{1/2})$，与二维平方自适应递归算法的最优性一致。本工作展示了多种方法，包括升序分解、奇数主序列与部分序列、T-级数、素数复合计数函数、$Z$-模形式以及组合方面。
英文摘要：
The prime-counting function $π(x)$ which returns the number of primes smaller or equal to a given number is a topic of interest in number theory. An algorithm based on a cyclic group isomorphic to $Z/nZ$, the so-called $Z$-functions, was proposed in view to outperform its pieers. The approach suggests a time complexity $\mathcal{O}(x^{1/2})$ in agreement with optimality of a 2-D squared adaptive-recursive algorithm. The present work is a presentation of various approaches as ascending factorization, the main sequence of odd numbers and partial sequences, T-series, counting function of prime composites, $Z$-modular forms and combinatorial aspects.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
