One more disproof for the Legendre’s conjecture regarding the prime counting function 𝜋(𝑥)

Reza Farhadian and Rafael Jakimczuk
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 3, Pages 84–91
DOI: 10.7546/nntdm.2018.24.3.84-91
Full paper (PDF, 187 Kb)

Details

Authors and affiliations

Reza Farhadian
Department of Statistics, Lorestan University
Khorramabad, Iran

Rafael Jakimczuk
División Matemática, Universidad Nacional de Luján
Buenos Aires, Argentina

Abstract

Let 𝜋(𝑥) denote the prime counting function, i.e., the number of primes not exceeding 𝑥. The Legendre’s conjecture regarding the prime counting function states that 𝜋(𝑥) = 𝑥 / (log 𝑥 − 𝐴(𝑥)), where Legendre conjectured that lim_𝑥→∞𝐴(𝑥) = 1.08366…, which is the Legendre’s constant. It is well-known that lim_𝑥→∞𝐴(𝑥) = 1, and hence the Legendre’s conjecture is not true. In this article we give various proofs of this limit and establish some generalizations.

Keywords

• Primes
• Prime counting function
• Legendre’s constant

2010 Mathematics Subject Classification

• Primary 11A41
• Secondary 11A25, 11N05

References

1. Chebyshev, P. L. (1851) Sur la fonction qui détermine la totalité des nombres premiers inférieurs à une limite donnée, Mémoires présentés à l’Académie Impériale des Sciences de St-Pétersbourg par divers Savants et lus dans ses Assemblées, Bd. 6, S., 141–157.
2. Cipolla, M. (1902) La determinazione assintotica dell’𝑛𝑖𝑚𝑜 numero primo, Rend. Acad. Sci. Fis. Mat. Napoli, Ser. 3, 8, 132–166.
3. Dusart, P. (1998) Sharper bounds for 𝜑, 𝜃, 𝜋 and 𝑝𝑛, Rapport de recherche, Université de Limoges.
4. Dusart, P. (1999) Inégalités explicites pour 𝜓(𝑋), 𝜃(𝑋), 𝜋(𝑋) et les nombres premiers, C. R. Math. Acad. Sci. Soc. R. Can., 21, 53–59.
5. Farhadian, R. (2017) On a Conjecture of Legendre, MAYFEB Journal of Mathematics, 3, 53–58.
6. Legendre, A. M. (1798) Essai sur la Théorie de Nombres, 1st ed., Paris: Duprat, p. 19.
7. Legendre, A. M. (1808) Essai sur la Théorie de Nombres, 2nd ed., Paris: Courcier, p. 394.
8. LeVeque, J. W. (1958) Topics in Number Theory, Volume II, Addison-Wesley.
9. Rey Pastor, J., Pi Calleja, P. & Trejo, C. (1969) Análisis Matemático, Editorial Kapelusz.