https://riemannhypothesis.blog/posts/ Recent content in Posts 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/2016/10/numbers-of-the-world/ Thu, 20 Oct 2016 16:41:00 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2016/10/numbers-of-the-world/ Recently Matt Parker uploaded a video to his YouTube channel where he discussed numbers and the words used to represent them in different languages, more precisely the length of these words: The basic idea is the following: one has 3 letters, two has 3 letters, three has 5 letters, four has 4 letters, five has 4 letters, six has 3 letters, seven has 5 letters, eight has 5 letters, nine has 4 letters, ten has 3 letters, and so on… This can be seen as a function https://riemannhypothesis.blog/2016/08/primes-from-a-different-world/ Fri, 05 Aug 2016 15:38:11 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2016/08/primes-from-a-different-world/ Every textbook on number theory will begin with a treatise on prime numbers; every treatise on prime numbers will begin by emphasising their importance as building blocks or atoms of our number system: every integer can be expressed as a product of prime numbers in one way and one way only. Six is two times three and there is no other way to decompose it.1 Euclid proved this over two thousand years ago and it is so fundamental (hence the name fundamental theorem of arithmetic) to our thinking about numbers that we take it for granted. 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/2015/10/a-toy-keyless-encryption-protocol/ Fri, 02 Oct 2015 11:00:37 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2015/10/a-toy-keyless-encryption-protocol/ Cryptography is a natural application of number theory and so I’d like to write down a few thoughts about it in this blog. (The fact that there are real world applications to number theory deserves some appreciation in itself, but it would throw us too much off track here.) One particularly nice feature of cryptography is the ability to explain its inner workings with real world analogies about security. For instance, one way two parties (who we, by convention, call Alice and Bob) could hide their secrete communication is if Alice writes a letter, puts it in a box, and locks it with a padlock for which both she and Bob have a key, but no one else. https://riemannhypothesis.blog/2015/09/prime-generating-sequences/ Wed, 23 Sep 2015 20:24:52 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2015/09/prime-generating-sequences/ A couple of months ago (really, a long, long time ago) I found an interesting question on Mathematics Stack Exchange (another site to effectively waste away hours of your life). It reminded me of my Bachelor’s thesis (which I wrote a really, really long time ago) about the sequence \[ g_n=\mathrm{gcd}(n,a_{n-1})=(n,a_{n-1}) \quad\text{for}\quad n>1, \] where \(a_1=7\) and \[ a_n=a_{n-1}+g_n. \] Here, \(\mathrm{gcd}(a,b)=(a,b)\) stands for the greatest common divisor,1 i.e., the largest integer \(d\) that divides both \(a\) and \(b\). https://riemannhypothesis.blog/2014/12/if-one-at-a-time-is-too-difficult-try-all-at-once/ Sat, 27 Dec 2014 12:00:16 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2014/12/if-one-at-a-time-is-too-difficult-try-all-at-once/ In the past months, I spent as much time as I had on taking online courses at Coursera. One particularly interesting course, both from a mathematical and computational point of view, is Analytic Combinatorics which applies combinatorics (i.e., the art of counting) to the analysis of algorithms by finding formulae, exact or asymptotic, for their running time. It is notoriously difficult to find exact formulae for general combinatorial constructs. Typically, we want to know how many objects, e. 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. https://riemannhypothesis.blog/2014/11/how-not-to-earn-a-million-dollars/ Sun, 02 Nov 2014 12:06:30 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2014/11/how-not-to-earn-a-million-dollars/ I recently spent some time on the formidable website Numberphile which explains mathematical ideas, some important, some recreational, in short and accessible videos. Definitely worth checking out. One of the videos that is most relevant to us explains the Riemann Hypothesis: As mentioned before, it’s not easy to explain the details and the beauty of the Riemann Hypothesis in few words, but I think the video definitely succeeds in compressing the essentials into 17 minutes. https://riemannhypothesis.blog/2014/10/tossing-the-prime-coin/ Sun, 12 Oct 2014 20:27:46 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2014/10/tossing-the-prime-coin/ One of the problems with explaining the Riemann Hypothesis is that its fascination comes from its deep connection to prime numbers, but its definition is in terms of complex analysis which requires a fair deal of undergraduate mathematics to understand – and that is before you even got started to grasp what the heck the zeta-zeros have to do with the distribution of primes. My “cocktail party explanation” of the Riemann Hypothesis would usually be something like: “The prime numbers are as equally distributed as you could wish for. https://riemannhypothesis.blog/2014/04/integral-madness/ Sun, 13 Apr 2014 15:25:54 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2014/04/integral-madness/ We’ve seen the calculus version \[ J(x)=\frac{1}{2\pi i}\int_{a-i\infty}^{a+i\infty}\log\zeta(s)x^s\frac{\mathrm{d}s}{s}, \] of the Euler product, and we know how to express \(\xi(s)\) as a product over its roots \[ \xi(s)=\xi(0)\prod_\varrho\left(1-\frac{s}{\varrho}\right), \] where \[ \xi(s) = \frac{1}{2} \pi^{-s/2} s(s-1) \Pi(s/2-1) \zeta(s) \newline = \pi^{-s/2} (s-1) \Pi(s/2) \zeta(s). \] High time we put everything together – the reward will be the long expected explicit formula for counting primes! First, let’s bring the two formulae for \(\xi(s)\) together and rearrange them such that we obtain a formula for \(\zeta(s)\): https://riemannhypothesis.blog/2014/04/from-zeta-to-j-and-back-and-yet-again-back/ Sun, 06 Apr 2014 14:13:42 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2014/04/from-zeta-to-j-and-back-and-yet-again-back/ We know a lot about the \(\zeta\) and \(\xi\)-functions, we’ve learnt all about the different prime counting functions, most notably \(J(x)\), so it’s high time we found a connection between the two. Probably not too surprisingly, the crucial link is our good friend, the Euler product \[ \zeta(s)=\prod_{p}(1-p^{-s})^{-1}. \] What we want to develop now is a version of this product that will suit us to find a formula that magically can count primes. https://riemannhypothesis.blog/2014/02/infinity-is-worth-no-more-than-112/ Sun, 23 Feb 2014 10:38:53 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2014/02/infinity-is-worth-no-more-than-112/ On 16 January 1913, a confused manuscript reached the famous mathematician G. H. Hardy in Cambridge. Other researchers have received similar letters before, and rejected it due to the seemingly incoherent formulae mixed with trivial mathematical results. Professional mathematicians are used to receiving manuscripts by amateurs who believe to have solved famous problems, but this particularly odd scribble caught the eye: \[ 1+2+3+4+5+6+\ldots+\infty=-\frac{1}{12} \] Did this amateur mathematician really think that the sum of all natural numbers, a value that will exceed any given boundary at some point, will wind up being a negative fraction? https://riemannhypothesis.blog/2014/01/counting-primes-functionally/ Sun, 12 Jan 2014 11:31:23 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2014/01/counting-primes-functionally/ After all this playing with the \(\zeta\)-function it is time to return to the overall objective of this whole exercise: counting prime numbers. The idea behind analytic number theory is that primes are unpredictable on the small scale, but actually surprising regular on the large scale. This is why we’ll look at certain functions that behave pretty erratically when we look at every single value, but become smooth and “easy” to calculate once we “zoom out” and consider the global properties, the so-called asymptotic. https://riemannhypothesis.blog/2013/12/more-symmetry-and-another-product/ Sun, 01 Dec 2013 21:29:17 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2013/12/more-symmetry-and-another-product/ We’ve seen that \(\zeta(s)\) satisfies the functional equation \[ \zeta(1-s)=2^{1-s}\pi^{-s}\cos(\pi s/2)\Pi(s-1)\zeta(s). \] (Well, it still needs to be proved, but let’s just assume it’s correct for now.) The goal of this post is an even more symmetrical form that will yield the function \(\xi(s)\) which we can develop into an incredibly useful product expression. On our wish list for \(\xi(s)\) we find three items: It’s an entire function, i.e., a function that’s holomorphic everywhere in \(\mathbb{C}\) without any poles. https://riemannhypothesis.blog/2013/11/are-primes-independent/ Sun, 24 Nov 2013 14:32:23 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2013/11/are-primes-independent/ The question may sound silly, but I hope it will become apparent that it’s very reasonable to ask. What we will examine here is the probabilistic interpretation of the prime distribution. So, essentially we ask: “What’s the probability that a randomly chosen number is prime?” Those familiar with some basic probability theory know the notion of independency in this context, so the question I’m basically interested in here is if the probability to find a prime is independent of the preceding or following numbers. https://riemannhypothesis.blog/2013/10/perfect-symmetry/ Thu, 31 Oct 2013 23:17:14 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2013/10/perfect-symmetry/ So far, we have seen how the Euler product links the \(\zeta\)-function to the prime numbers. More precisely, it encodes the fundamental theorem of arithmetic. One may also say, it’s the analytic version of it, in a sense that should become clearer shortly. What we have done so far works perfectly for the real numbers. The sum \(\sum n^{-s}\) that defines \(\zeta(s)\) converges for \(s>1\), that’s how Leonhard Euler found his product, and that’s what Peter Gustav Lejeune Dirichlet used to prove the prime number theorem in arithmetic progressions. https://riemannhypothesis.blog/2013/10/does-the-euler-product-converge/ Tue, 08 Oct 2013 18:36:28 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2013/10/does-the-euler-product-converge/ Usually, I don’t care too much about convergence as a general overview of the argument is what I aim at here, and otherwise I’ll just trust that things “behave well”. But some words concerning convergence won’t harm. It’s a well known fact that the harmonic series (which we shortly touched in the previous post) \(\sum1/n\) diverges. I think the best (though not easiest) proof of this to compare it with the corresponding integral: https://riemannhypothesis.blog/2013/10/euler-product-revisited/ Sun, 06 Oct 2013 22:24:34 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2013/10/euler-product-revisited/ From the previous post we know that the harmless looking series \(\sum n^{-s}\) can be extended to the product \(\prod (1-p^{-s})^{-1}\). At first sight, this does not seem terribly helpful, and it actually makes the rather easy series more complicated. So what’s the big deal? It’s what has actually been suppressed in the above notation: The sequences we run through. The series runs over all natural numbers (or positive integers, if you prefer), the product runs through all prime numbers. https://riemannhypothesis.blog/2013/09/in-the-beginning-there-was-eulers-formula/ Sun, 29 Sep 2013 17:39:48 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2013/09/in-the-beginning-there-was-eulers-formula/ I will start this blog the way Bernhard Riemann started his paper: with Euler’s product formula that John Derbyshire called the golden key: \[ \zeta(s)=\sum_{n\ge1}n^{-s}=\prod_{p}(1-p^{-s})^{-1} \] This holds for any complex number \(s\) with \(\Re s > 1\). If you look up a proof in any modern textbook, you will find a number technical rearrangements that end up in an examination of the absolute convergence on both sides. But actually, the formula is nothing but a fancy way of writing out the Sieve of Eratosthenes.