https://riemannhypothesis.blog/tags/approximation/ Recent content in Approximation on Understanding the Riemann Hypothesis Hugo -- gohugo.io en markus@recommend.games (Markus Shepherd) markus@recommend.games (Markus Shepherd) Sun, 23 Nov 2014 10:47:17 +0000 https://riemannhypothesis.blog/2014/11/applying-the-explicit-formula/ Sun, 23 Nov 2014 10:47:17 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2014/11/applying-the-explicit-formula/ It’s quite some time since we arrived at Riemann’s main result, the explicit formula \[ J(x)=\mathrm{Li}(x)-\sum_{\Im\varrho>0}\left(\mathrm{Li}(x^\varrho)+\mathrm{Li}(x^{1-\varrho})\right)+\int_x^\infty\frac{\mathrm{d}t}{t(t^2-1)\log t}-\log2, \] where \(J(x)\) is the prime power counting function introduced even earlier. It’s high time we applied this! First, let’s take a look at \(J(x)\) when calculating it exactly: You see how this jumps by one unit at prime values (\(2\), \(3\), \(5\), \(7\), \(11\), \(13\), \(17\), \(19\)), by half a unit at squares of primes (\(4\), \(9\)), by a third at cubes (\(8\)), and by a quarter at fourth powers (\(16\)), but is constant otherwise.