Proof in the literature of an equality involving the prime counting function - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T11:46:49Z http://mathoverflow.net/feeds/question/38581 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38581/proof-in-the-literature-of-an-equality-involving-the-prime-counting-function Proof in the literature of an equality involving the prime counting function alext87 2010-09-13T14:17:51Z 2010-09-14T04:27:23Z <p>Let $$R(x) = \sum_{k=1}^{\infty}\frac{\mu(k)}{k}li(x^{1/k})$$ where $\mu$ is the Mobius function and $$li(x) = \int_0^x \frac{dt}{\log t}$$ Is there a proof in the literature of $$\pi(x)=R(x)-\sum_{\rho}R(x^{\rho})$$ where $\pi$ is prime counting function and the sum is over all complex zeros of $\zeta(s)$. The literature seems to treat it as fact while stating no proof is available - a strange situation.</p> <p>Thanks in advance. </p> http://mathoverflow.net/questions/38581/proof-in-the-literature-of-an-equality-involving-the-prime-counting-function/38595#38595 Answer by Micah Milinovich for Proof in the literature of an equality involving the prime counting function Micah Milinovich 2010-09-13T16:04:08Z 2010-09-13T16:04:08Z <p>It may be useful to read Section 10 of Chapter V of Ingham's "The Distribution of Prime Numbers."</p> <p>Let $\Pi(x)=\pi(x)+\frac{1}{2}\pi(x^{1/2})+...$, then Moebius proved that</p> <p>$$\pi(x) = \sum_{n=1}^\infty \frac{\mu(n)}{n} \Pi(x^{1/n}).$$</p> <p>However, this isn't overly illuminating because it shows that </p> <p>$$\pi(x) = \Pi(x) + O(\sqrt{x}/\log x ).$$</p> <p>and Littlewood showed that </p> <p>$$\Pi(x) - \ell i(x) = \Omega_\pm(\sqrt{x} \log\log\log x/\log x).$$ </p> http://mathoverflow.net/questions/38581/proof-in-the-literature-of-an-equality-involving-the-prime-counting-function/38646#38646 Answer by Gerry Myerson for Proof in the literature of an equality involving the prime counting function Gerry Myerson 2010-09-14T04:27:23Z 2010-09-14T04:27:23Z <p>Stopple, A Primer of Analytic Number Theory, proves a theorem which looks something like the one under discussion. On page 248, he has $$\pi(x)=R(x)+\sum_{\rho}R(x^{\rho})+\sum_{n=1}^{\infty}{\mu(n)\over n}\int_{x^{1/n}}^{\infty}{dt\over t(t^2-1)\log t}$$</p> <p>You say that the literature treats your formula as a fact, but you give no citation. Where in the literature do you find your formula? </p>