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
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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 lim𝑥→∞𝐴(𝑥) = 1.08366…, which is the Legendre’s constant. It is well-known that lim𝑥→∞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

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  5. Farhadian, R. (2017) On a Conjecture of Legendre, MAYFEB Journal of Mathematics, 3, 53–58.
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Cite this paper

APA

Farhadian, R. & Jakimczuk, R. (2018). One more disproof for the Legendre’s conjecture regarding the prime counting function 𝜋(𝑥). Notes on Number Theory and Discrete Mathematics, 24(3), 84-91, doi: 10.7546/nntdm.2018.24.3.84-91.

Chicago

Farhadian, Reza and Rafael Jakimczuk. “One More Disproof for the Legendre’s Conjecture Regarding the Prime Counting Function 𝜋(𝑥).” Notes on Number Theory and Discrete Mathematics 24, no. 3 (2018): 84-91, doi: 10.7546/nntdm.2018.24.3.84-91.

MLA

Farhadian, Reza and Rafael Jakimczuk. “One More Disproof for the Legendre’s Conjecture Regarding the Prime Counting Function 𝜋(𝑥).” Notes on Number Theory and Discrete Mathematics 24.3 (2018): 84-91. Print, doi: 10.7546/nntdm.2018.24.3.84-91.

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