Mertens first theorem states that $ \sum_{ p \le x } \frac{\log p}{p} = \log x + R $ with $| R | \le 2$ .
Is it correct that the limit $ \lim_{x \to \infty} \sum_{ p \le x } \frac{\log p}{p} - \log x $ exists? And if so, can somebody be so kind to post a link where I can find online a proof of the existence of this limit?
Thank you.
We can use the Prime Number Theorem in this form $$\theta\left(x\right)=\sum_{p\leq x}\log\left(p\right)=x+O\left(\frac{x}{\log^{2}\left(x\right)}\right) $$ to get by partial summation $$\sum_{p\leq x}\frac{\log\left(p\right)}{p}=\frac{\theta\left(x\right)}{x}+\int_{2}^{x}\frac{\theta\left(t\right)}{t^{2}}dt=\log\left(x\right)+R+o\left(1\right) $$ then $$ \limsup_{x\rightarrow\infty}\left(\sum_{p\leq x}\frac{\log\left(p\right)}{p}-\log\left(x\right)\right)=\liminf_{x\rightarrow\infty}\left(\sum_{p\leq x}\frac{\log\left(p\right)}{p}-\log\left(x\right)\right)=R $$ and this prove the existence of the limit.