Asymptotic formula for the sum of log(p)/p

[A. Bahat]

I know that $$\sum_{p\leq n}\frac{\log p}{p}=\log n + O(1),$$ where $p$ ranges over primes, can be proved using the Prime Number Theorem. However, I was under the impression (which may very well be wrong) that this result is not nearly as deep as PNT and can be proved without it. I ask because I came across an intriguing proof that $$\sum_{p\leq n}\frac{1}{p}=\log\log n + M + O\left(\frac{1}{\log n}\right)\text{ for some constant }M,$$ using Abel's summation formula to get the error bound, but it depends on this result and applies it without proof. Does anyone know how to prove the first equation without appealing to PNT?

 

[Petek]

Your first equation is Theorem 8.8 (b) in An Introduction to the Theory of Numbers by Niven, Zuckerman and Montgomery. That text doesn't contain a proof of the PNT, so it probably is what you're looking for.

 

[A. Bahat]

Thank you, that's just what I needed.