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.
Proving $\sum\limits_{k=1}^{\pi(n)-1} [ \theta(p_k) (1/{p_k}-1/p_{k+1})] -\ln(n)$ converges
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number-theory
prime-numbers
analytic-number-theory
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2Why? 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? – 2011-06-05