## Concerning the sequence ∑p≤n,p⌊lognlogp⌋/n \sum_{p \le n , p} \big\lfloor\frac{\log n}{\log p}\bigr\rfloor/n where p is prime

The sequence (∑p≤n,pisprime⏟⌊lognlogp⌋n)∞n=1 is bounded , my question is what is the lim sup of this sequence ? Is the sequence convergent ? If it does , what is the limit ? Answer We have1n∑p≤n⌊log(n)log(p)⌋=1n∑p≤n|{m∈N+:pm≤n}|=1n∑pm≤n1=π(n)n+O(log(n)√n)=O(1log(n)+log(n)√n)=O(log(n)√n). where π(n)=∑p≤n1. Expanding a little bit, note that 1)|{m∈N+:pm≤x}|=|{m∈N+:mlog(p)≤log(x)}|=⌊log(x)log(p)⌋ 2)We have the bound |{p:pm≤x,m≠1}|=O(√xlog(x)) 3)By Prime Number Theorem π(x)=(xlog(x)). AttributionSource : Link … Read more