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An open problem in analytic number theory is to determine whether or not the following infinite series converges:

$\sum_{n=2}^{\infty} \frac {(-1)^{\pi(n)}}{n log n}$

Here, $\pi(n)$ is the prime counting function so the problem reduces to understanding the parity of this function. I suspect that the sum converges after some numerical calculations but I don't know what type of theorem needs to be established to prove convergence. I posted the problem on Terry Tao's blog and he tells me that some kind of equidistribution theorem is needed but I don't have any idea how to even formulate it. Does any one have any ideas?

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  • $\begingroup$ Why is this problem interesting? That seems like a fairly arbitrary expression to me, why should we care if it converges? Maybe you could link to where this problem originates, to give us a better feel for why this is interesting. $\endgroup$
    – Jacques Carette
    Commented Nov 4, 2011 at 11:58
  • $\begingroup$ I suspect that convergence is tantamount to finitely many small gaps between primes, which would have an effect on a number of other conjectures. Is that part of the motivation for investigating this? Have you considered the relationship of this sum to those and other conjectures in number theory? Gerhard "Ask Me About System Design" Paseman, 2011.11.04 $\endgroup$
    – Gerhard Paseman
    Commented Nov 4, 2011 at 12:35
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    $\begingroup$ Any references? $\endgroup$
    – Igor Rivin
    Commented Nov 4, 2011 at 13:27

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