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

Yuri Heymann

Abstract

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.

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

Abstract

The prime-counting function 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 , the so-called -functions, was proposed in view to outperform its pieers. The approach suggests a time complexity 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, -modular forms and combinatorial aspects.
Paper Structure (4 sections, 15 equations)

This paper contains 4 sections, 15 equations.

Table of Contents

  1. Introduction
  2. The sequence $\{S_n\}$ a key concept for primes; definitions and propositions
  3. Ascending factorization and counting of primes composites
  4. $Z$-functions as bricks for prime composites and factorization