Asymptotic and non-asymptotic bounds of π (x) Madieyna Diouf 1 Abstract: Observations obtained from researches on a modified sieve of Eratosthenes, high- light a function of the form x log X that gives a better approximation of π(x) than x log x . We present S π (x)= x log X and its advantages over the Tchebycheff’s bounds of π(x). Keywords: prime, prime counting function, bounds. 2010 Mathematics Subject Classification: 11N05. 1 Introduction The approximation π(x) ∼ x log x (1) is known to us as the prime number theorem. It started with predictions by Legendre and Gauss [1], then a first real progress toward a proof by Tchebycheff [4], and finally a complete proof in 1896 by Hadamard and independently de la Valle Poussin. The prime number theorem is indisputably the key stone of number theory, and attempts to give better approximations of π(x) have continued in view of results by many authors. The best estimations among these results are namely, that of Rosser [2]: for x> 55 x log x +2 <π(x) < x log x - 4 . (2) Then by Dusart [3] expressed in the following. The first inequality from the left of (3) holds for all x ≥ 599 and the second one for x ≥ 355991. x log x  1+ 1 log x  <π(x) < x log x  1+ 1 log x + 2.51 log 2 x  . (3) In this short note, we prove the following bounds of π(x), where X shall be further defined. Asymptotic x log x ∼ x log X ∼ π(x). (4) Non-asymptotic x log x <x log X<π(x). (5) Relation (5) shows that x log X is clearly an improvement to what is known in the literature as Tchebycheff’s bounds of the number of primes less than x, and it absolutely represents an alternative to x/ log x. One can see that the function x log X is not derived from x/ log x as previous authors succeeding Tchebycheff have obtained their lower bounds. Furthermore, our 1