Precise relation between prime number theorem and zero-free region - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T00:20:15Z http://mathoverflow.net/feeds/question/25789 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25789/precise-relation-between-prime-number-theorem-and-zero-free-region Precise relation between prime number theorem and zero-free region Akela 2010-05-24T17:44:39Z 2010-05-24T18:20:55Z <p>I was wondering about the following, and I was hoping that some expert here could answer, rather than me indulging in a search for a needle in the haystack of formulas in books like Titchmarsch.</p> <p>Notation:</p> <ul> <li>$\zeta(s)$ is the Riemann zeta function.</li> <li>$f : \mathbb R^+ \rightarrow (0,1/2)$ is such that $\zeta(s)$ does not vanish between $s = 1+it$ and $s=1 - f(t) + it$.</li> <li>$\pi(x)$, $Li(x)$ as in <a href="http://en.wikipedia.org/wiki/Prime_number_theorem" rel="nofollow">wikipedia</a>.</li> </ul> <p>Assuming the above data, suppose the version of the prime number theorem that can be proven is:</p> <p>$$\pi(x) = Li(x) + O\left(G(x)\right)$$</p> <p>Question:</p> <blockquote> <p>Can G(x) be given a closed form expression showing its precise(if and only if) dependence on $f(t)$?</p> </blockquote> <p>Heuristics: When $f = 0$, $G(x) = x \mathrm{e}^{-a\sqrt{\ln x}}$ and when $f = 1/2$, $G(x) = \sqrt x \ln x$. So possibly there would be a term like $x^{1-f(x)}$ in a putative expression for $G(x)$.</p> http://mathoverflow.net/questions/25789/precise-relation-between-prime-number-theorem-and-zero-free-region/25790#25790 Answer by David Hansen for Precise relation between prime number theorem and zero-free region David Hansen 2010-05-24T18:03:02Z 2010-05-24T18:20:55Z <p>Your heuristic is wrong: $G(x)=x\exp{(-a\sqrt{\log{x}}})$ follows from $f=\frac{c}{\log{(|t|+3)}}$ for some fixed real $c>0$.</p> <p>I really don't want to tell you the answer, because this is a great exercise! A big hint: use the "approximate explicit formula"</p> <p>$\psi(x)=x-\sum_{|\rho|\leq T} \frac{x^{\rho}}{\rho}+O(T^{-1} x \log^2{x}),$ </p> <p>bound the sum over zeros trivially given what you know about $f$, and then choose $T$ so that the two error terms balance.</p>