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

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.

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Why? Is this your homework? Is it something you read somewhere? Is it a conjecture of yours? Are we allowed to use the Prime Number Theorem? Is this just an exercise in summation by parts? – Gerry Myerson Jun 5 '11 at 13:10

## 1 Answer

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,

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