An interesting series involving the prime counting function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T17:59:25Z http://mathoverflow.net/feeds/question/80023 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80023/an-interesting-series-involving-the-prime-counting-function An interesting series involving the prime counting function Mustafa Said 2011-11-04T09:15:54Z 2011-11-04T09:15:54Z <p>An open problem in analytic number theory is to determine whether or not the following infinite series converges:</p> <p>$\sum_{n=2}^{\infty} \frac {(-1)^{\pi(n)}}{n log n}$</p> <p>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?</p>