p-composites of the main sequence of odd numbers as building blocks for $\pi(x)$

Abstract

The prime-counting function $\pi(x)$ 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.