There are some interesting statements that are equivalent to the Riemann Hypothesis. What "equivalent" means is that if the statement is true then RH must be true, and if RH is true then the statement must be true.<p>Here's one I find particularly nice.<p>Let s(n) = the sum of the divisors of n, for a positive integer n. For example s(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28.<p>Let H(n) = 1 + 1/2 + 1/3 + ... + 1/n.<p>The RH is equivalent to the claim that for every integer n >= 1:<p>s(n) <= H(n) + exp(H(n)) log(H(n))<p>This is due to Jeffry C. Lagarias. Here's his paper showing the equivalence: <a href="https://arxiv.org/abs/math/0008177" rel="nofollow">https://arxiv.org/abs/math/0008177</a>
3Blue1Brown recently did a video on the Riemann zeta function: <a href="https://www.youtube.com/watch?v=sD0NjbwqlYw" rel="nofollow">https://www.youtube.com/watch?v=sD0NjbwqlYw</a><p>It's not as in depth but it has some helpful visualisations that I've never seen elsewhere.
Shameless promotion: I published a book recently trying to explain RH, and my coauthor gave a great talk about it, which is here: <a href="http://wstein.org/rh" rel="nofollow">http://wstein.org/rh</a>
Great article!<p>One of the things about the Riemann Hypothesis is that the search space for a proof is more wide than it is deep. For each given idea or approach it doesn't take relatively long to get to the forefront of what is known, and an expert can often tell you right from the beginning that the whole class of approaches may not work due to some known phenomena, or that they would have to involve certain complications to not pick up on various almost-counterexamples. Furthermore, in these 150 years not only has the right path/approach not been found, but there isn't a truly compelling reason to believe that RH is true, aside from numerical evidence and a belief in beauty.<p>I think it'd be fascinating to put together an online resource to organize the possible approaches, list the knowledge prerequisites, show the potential counterexamples and stumbling blocks to each approach. This would help anyone interested in the problem, and once it is sufficiently developed it would allow non-specialists to contribute productively. This could be LaTeX on github, it could be more of a traditional wiki, but now I really want to get this going.
This is a great introduction to analytic number theory.<p>I was expecting the standard dumbed down piece with no mathematical insight and far-fetched analogies. Instead, the post is mathematically rigorous and deep, and yet it manages to be completely self-contained and clear. Amazing work.
[nitpicking]<p>This version of Euclid's proof does not look like the one on Wikipedia, which is not a proof by contradiction. The article[0] mentions that:<p>"Euclid is often erroneously reported to have proved this result by contradiction, beginning with the assumption that the finite set initially considered contains all prime numbers, or that it contains precisely the n smallest primes, rather than any arbitrary finite set of primes."<p>[/nitpicking]<p>[0]: <a href="https://en.wikipedia.org/wiki/Euclid's_theorem#Euclid.27s_proof" rel="nofollow">https://en.wikipedia.org/wiki/Euclid's_theorem#Euclid.27s_pr...</a>
Hm, except for using Gamma(n) = (n-1)! instead of Pi(n) = n!, this presentation seems to follow the notation of this book:<p><a href="https://books.google.ca/books/about/Riemann_s_Zeta_Function.html?id=ruVmGFPwNhQC" rel="nofollow">https://books.google.ca/books/about/Riemann_s_Zeta_Function....</a><p>This is most obvious in the naming of the J function near the bottom, which I have not seen anywhere else. Riemann originally called that function a generic f.<p>At any rate, Edwards' book is great because it develops the theory from a historical viewpoint. It begins with a very well-annotated exposition of Riemann's original paper and the rest of the book goes on to explain other mathematicians' successive results in filling in all of the gaps that Riemann left in his paper. I recommend this book to any serious student of zeta.
For those who did not read it or missed it, at the end of the article there is a link to Jørgen Veisdal's 2013 undergraduate thesis paper.<p><a href="http://www.jorgenveisdal.com/files/jorgenveisdal-thesis13.pdf" rel="nofollow">http://www.jorgenveisdal.com/files/jorgenveisdal-thesis13.pd...</a>
I'm pretty awful at math and I found this explanation to be wonderful despite many parts of it going over my head. My brain naturally despises numbers (dyslexia)but your textual descriptions of formulas seemed to help me bridge the gap better than most texts.
here is a more technical discussion by Paul Bourgade @ NYU. I still re-read it from time to time for orientation.<p>Quantum chaos, random matrix theory, and the Riemann ζ function<p><a href="http://www.cims.nyu.edu/~bourgade/papers/PoincareSeminar.pdf" rel="nofollow">http://www.cims.nyu.edu/~bourgade/papers/PoincareSeminar.pdf</a>
I went to a talk on the Reimann hypothesis with a physicist friend. The chair of our math department gave the talk. At one point he stated "Do you know why the Reimann hypothesis is important?, it's because it's the Reimann hypothesis". Suffice to say the entire lecture was waaaay over our heads.
> The gamma function Γ(z) is defined for all complex values of z larger than zero<p>What does "larger than zero" mean for complex numbers?
For the curious, much of this is covered in depth in a graduate analytic number theory first course.<p>As fascinating as the Riemann Zeta Hypothesis is, I think something almost as fascinating is the works of Ramanujan - to this day, the genius Indian's work is studied in earnest, and amongst the most famous number theory work to this day.
This article is about math.<p>In my ignorance, I mistook the article to be about the monitoring project[1] -- I expected some kind of proof/theorem behind the monitoring project.<p>This article is not about the monitoring project.<p>[1]: <a href="http://riemann.io/" rel="nofollow">http://riemann.io/</a>
Great post, I have been reading <a href="https://www.amazon.com/Riemann-Hypothesis-Greatest-Unsolved-Mathematics/dp/0374529353" rel="nofollow">https://www.amazon.com/Riemann-Hypothesis-Greatest-Unsolved-...</a> for over 10 years now ;-)
> The real valued zeta function is given for r and n, two real numbers<p>followed by a function of one real variable with an infinite sum of expressions containing integer r.<p>Sorry, but little things like these completely put me off in math articles.
<i></i><i>META</i><i></i><p>Why are people still posting things on Medium?<p>It's a platform that the founder knows has absolutely no clue about how to sustain it. After 5 years, and over hundred million dollars of investments.<p><i></i><i>end meta</i><i></i><p>This is a good post, and it should be put somewhere that will keep it.<p>That said, "RH" stuff now:<p>Fuck Riemann. An entire school of music theory that has almost nothing to do with him is named after him. It's a ginormoulsy idiotic way to understand music.<p>The people who think of him as a great thinker as it applies to music are completely out of their minds.<p>/trolling, sort of. a little.