https://riemannhypothesis.blog/tags/zeta-function/ Recent content in Zeta function on Understanding the Riemann Hypothesis Hugo -- gohugo.io en markus@recommend.games (Markus Shepherd) markus@recommend.games (Markus Shepherd) Wed, 31 May 2017 05:41:25 +0000 https://riemannhypothesis.blog/2017/05/the-prime-bet/ Wed, 31 May 2017 05:41:25 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2017/05/the-prime-bet/ Let’s say you sit in a pub, minding your own business, when all of a sudden a stranger walks up to you and offers you a bet: We’ll choose two positive integers at random. If they have any divisor in common (other than \(1\)) I’ll pay you a dollar, else you’ll pay me a dollar. Are you in? Apart from the question what kind of establishments you frequent, you should be wondering: is this a good bet for you? https://riemannhypothesis.blog/2016/04/visualising-the-riemann-hypothesis/ Sun, 10 Apr 2016 18:59:25 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2016/04/visualising-the-riemann-hypothesis/ Update 2024-10-20: Watch the video now in 4K! One stubborn source of frustration when working with complex numbers is the fact that visualisation becomes tedious, if not impossible. Complex numbers have 2 “real” dimensions in themselves, which give rise to the complex plane. That’s all good and fair. But if you talk about a function with complex domain and codomain, you already deal with a 4-dimensional graph. Unfortunately, my mind can only handle 3 dimensions (on a good day). 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.