On the Function π(x)

Abstract

Abstract Let π(x) be the number of primes not exceeding x. We prove that π ( x ) < x log x - 1.006789 for x ≥ e 10 12 , and that for sufficiently large x : x log x - 1 + ( log x ) - 1.5 + 2 ( log x ) - 0.5 < π ( x ) < 1 log x - 1 - 2 ( log x ) - 0.5 - ( log x ) - 1.5 . We finally prove that for x ≥ e 10 12 and k = 2, 3,…, 147297098200000, the closed interval [(k – 1)x, kx] contains at least one prime number, i.e. the Bertrand's postulate holds for x and k as above.