New estimations for numerical analysis approach to twin primes conjecture

Gabriele Di Pietro
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 3, Pages 580–586
DOI: 10.7546/nntdm.2024.30.3.580-586
Full paper (PDF, 195 Kb)

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Authors and affiliations

Gabriele Di Pietro
Via delle Ville, 18, 64026, Roseto degli Abruzzi (TE), Italy

Abstract

This paper provides a better approximation of the functions presented in the article “Numerical Analysis Approach to Twin Primes Conjecture” (see [3]). The new estimates highlight the approximations used in the previous article and the validity of Theorems 1 and 2 through the use of the false hypothesis based on the distribution of primes punctually following the Logarithmic Integral (see [4] and [7], pp. 174–176) will be re-evaluated.

Keywords

• Numerical analysis
• Number theory
• Sieves

2020 Mathematics Subject Classification

• 11N35

References

1. De la Vallée Poussin, C. J. (1896). Recherches analytiques la théorie des nombres premiers. Annales de la Société scientifique de Bruxelles, 20, 183–256.
2. Derbyshire, J. (2004). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin.
3. Di Pietro, G. (2021). Numerical analysis approach to twin primes conjecture. Notes on Number Theory and Discrete Mathematics, 27(3), 175–183.
4. Gauss, C. F. (1863). Werke, Band 10, Teil I. p. 10.
5. Hadamard, J. (1896). Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques. Bulletin de la Societe Mathematique de France, 24, 199–220.
6. Hardy, G. H., & Wright, E. M. (1979). An Introduction to the Theory of Numbers (5th ed.). Oxford Science Publications, Oxford, England: Clarendon Press, pp. 354–358.
7. Havil, J., & Dyson, F. (2003). Problems with Primes. In Gamma: Exploring Euler’s Constant. Princeton; Oxford: Princeton University Press, pp. 163–188.
8. Rosser, J. B., & Schoenfeld, L. (1962). Approximate formulas for some functions of prime numbers. Illinois Journal of Mathematics, 6(1), 64–94.
9. Selberg, A. (1949). An elementary proof of the Prime-Number Theorem. Annals of Mathematics, 50(2), 305–313

Manuscript history

• Received: 27 February 2024
• Revised: 3 October 2024
• Accepted: 9 October 2024
• Online First: 11 October 2024

Copyright information

Ⓒ 2024 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).