TITLE: Equivalence to the prime number theorem
QUESTION [11 upvotes]: I was just reading this question and answer:   How will this equation imply PNT 
and it raised a whole new question:

Given that  $\sum_{n\le x} \Lambda(n)=x+o(x)$, prove that $$\sum_{n\le x} \frac{\Lambda(n)}{n}=\log x-\gamma +o(1),$$ where gamma is the Euler constant. 

The above question is for the bounty.  The full version above has been bothering me.  Gerry Myerson pointed out we can prove $\sum_{n\leq x } \frac{\Lambda(n)}{n} =\log x +O(1)$ from only chebyshev estimate.  But this is not my question.
My Attempts: I tried partial summation I end up with something like $\sum_{n} \frac{\psi(n)-n}{n^2}$.  for that to converge $o(n)$ isn't strong enough.  To do it my way, I would need to assume $$\psi(x)-x = O\left(\frac{x}{(\log x)^{1+\epsilon}}\right)$$ so that it converges.  (Otherwise it could be as big as $\log x$ which is no good)
Can we prove the above estimate using only the basic prime number theorem $\psi(x)-x=o(x)$?  Why or why not?  Thank you!
Please note that my question is about a subtlety with converge, and blunt partial summation doesn't seem to work.

REPLY [1 votes]: By definition
$$\gamma=\sum_{n\le x}{1\over n}-\log x+o(1).$$
So we see that $$\log x +\gamma + o(1)=\sum_{n\le x}{1\over n}.$$
So it is sufficient to show that $$\sum_{n\le x}{\Lambda(n)-1\over n}=-2\gamma+o(1).$$
Then using Möbius inversion in the form
$$G(x)=\sum_{n\le x}F\left({x\over n}\right)\iff F(x)=\sum_{n\le x}\mu(n)G\left({x\over n}\right)$$
and the asymptotic estimate
$$\sum_{n\le x}\bigg(\log n-d(n)\bigg)=-2\gamma x+O(\sqrt{x})\quad (*)$$
we get the desired result. Note:  you'll end up using the PNT in a $\mu$ function estimate during the Möbius inversion, this is how you use your given, and the given estimate is where you save over just using the PNT by itself. As @Phira notes:  this is impossible.
Note:  $(*)$ can be proved pretty easily using standard summation techniques, the first term is $\log[x]!=x\log x -x+O(\log x)$ which is just partial summation and the second is $x\log x +(2\gamma-1)x+O(\sqrt{x})$ by using Dirichlet's hyperbola method.
{"set_name": "stack_exchange", "score": 11, "question_id": 76778}