p-composites of the main sequence of odd numbers as building blocks for $π(x)$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866929277487808512 |
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| author | Heymann, Yuri |
| author_facet | Heymann, Yuri |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09642 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | p-composites of the main sequence of odd numbers as building blocks for $π(x)$ Heymann, Yuri General Mathematics 11A41, 11B50 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. |
| title | p-composites of the main sequence of odd numbers as building blocks for $π(x)$ |
| topic | General Mathematics 11A41, 11B50 |
| url | https://arxiv.org/abs/2403.09642 |