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

Abstract

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

Locations

• Notes on Number Theory and Discrete Mathematics - View - PDF