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 primecounting function assert that (2x) < 2(x) and (x)2 < e xlog x ( xe ), respec-tively, for suciently large x. in this paper we give an asymptotic expansionfor ( x) as the common key to study landau's inequality and ramanujan'sinequality. then, we give several renements and generalizations of theseinequalities.