On a Weighted Asymptotic Expansion Concerning Prime Counting Function and Applications to Landau s and Ramanujan s Inequalities

Landau s inequality and Ramanujan s inequality concerning prime counting function assert that (2x) 2 (x) and (x)2 e x log x ( x e ), respec- tively, for su ciently large x. In this paper we give an asymptotic expansion for ( x) as the common key to study Landau s inequality and Ramanujan s inequality. Then, we give several re nements and generalizations of these inequalities.