Estimate needed on a sum involving the prime counting function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T00:31:50Z http://mathoverflow.net/feeds/question/115580 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115580/estimate-needed-on-a-sum-involving-the-prime-counting-function Estimate needed on a sum involving the prime counting function Mustafa Said 2012-12-06T06:50:34Z 2012-12-06T06:50:34Z <p>Let $F(x) = \sum_{1 &lt; n \leq x} (-1)^{\pi (n)}$ where $\pi (n)$ is the prime counting function.</p> <p>I am trying to understand how $F(x)$ behaves as $x \to \infty$. In particular, what are the asymptotics for $F$? I can't seem to use any tools that handle sums of the form, $\sum_{n \leq x} f(n)$ where $f:\mathbb{N} \to \mathbb{C}$. Does anyone have any suggestions?</p>