On the Function π(x)

Let π(x) be the number of primes not exceeding x. We prove that π(x)<xlogx-1.006789 for x≥e1012, and that for sufficiently large x:xlogx-1+(logx)-1.5+2(logx)-0.5<π(x)<1logx-1-2(logx)-0.5-(logx)-1.5. We finally prove that for x≥e1012 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.