Proving $\sum\limits_{k=1}^{\pi(n)-1} [ \theta(p_k) (1/{p_k}-1/p_{k+1})] -\ln(n)$ converges

Question

Prove the sequence $a_n$ defined by $a_n = \sum\limits_{k=1}^{\pi(n)-1} [ \theta(p_k) (1/{p_k}-1/p_{k+1})] -\ln(n)$ converges, where $p_k$ denotes the $k$-th prime and $\vartheta(x)$ is Chebyshev's theta function.

Answer 1

Hint:

Apply summation by parts. Then you will get something which looks like $$\sum_{p\leq x} \frac{\log p}{p}.$$ This sum is equal to $$\log x +C+O\left(e^{-c\sqrt{\log x}}\right)$$ using partial summation an the quantitative prime number theorem.

Without the prime number theorem, you can show that the sequence is bounded by some constant, but it is unlikely that you can prove it has a limit.

Hope that helps,