The Riemann Hypothesis, explained
medium.com
medium.com
Here's one I find particularly nice.
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.
Let H(n) = 1 + 1/2 + 1/3 + ... + 1/n.
The RH is equivalent to the claim that for every integer n >= 1:
s(n) <= H(n) + exp(H(n)) log(H(n))
This is due to Jeffry C. Lagarias. Here's his paper showing the equivalence: https://arxiv.org/abs/math/0008177
It's not as in depth but it has some helpful visualisations that I've never seen elsewhere.
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.
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.
You wouldn't consider the analogue of the Riemann hypothesis for the zeta function of a smooth projective variety over a finite field, proven by Deligne, to be a compelling reason to believe the RH is true?
Numerical evidence isn't proof, and yes I know about Littlewood and Skewes, but it starts to get suggestive.
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.
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:
"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."
[/nitpicking]
[0]: https://en.wikipedia.org/wiki/Euclid's_theorem#Euclid.27s_pr...
Euclid's proof: Every finite set of primes {p1,p2,...pn} has a prime, namely p1p2...pn + 1, not in the set.
Contradiction proof: The hypothetical finite set of all primes {p1,p2,...pn} would have a prime, namely p1p2...pn + 1, not in the set.
I'd hardly call the latter a different proof.
2*3*5*7*11*13 + 1 = 30031 = 59*509
It is trivial to show that there are infinitely many primes from it though through proof by contradiction as follows:
1. Assume there are finitely many primes. 2. Write a list of all the primes. (Writing a list of finite elements is possible) 3. There exists a prime not in that list (By Euclid's proof) 4. All primes are in the list.
3 and 4 contradict and therefore the single assumption is incorrect. So there are infinitely many primes.
As a sidenote, it seems that proving there are infinitely elements of any set will need to be proven by contradiction. Is there any proof of that?
https://books.google.ca/books/about/Riemann_s_Zeta_Function....
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.
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.
1. Men of Mathematics (E.T. Bell, 1937)
2. The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike (Peter Borwein et.al, 2007)
3. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (John Derbyshire, 2004)
4. Unknown Quantity: A Real and Imaginary History of Algebra (John Derbyshire, 2007)
5. Journey through Genius: The Great Theorems of Mathematics (William Dunham, 1991)
6. Riemann’s Zeta Function (H. M. Edwards, 1971)
7. Gamma: Exploring Euler’s Constant (Julian Havil, 2009)
8. The Man Who Loved Only Numbers (Paul Hoffman, 1999)
9. The Riemann Zeta-Function: Theory and Applications. (Alexander Ivic, 2004)
10. e: The Story of a Number (Eli Maor, 2009)
11. An Imaginary Tale: The Story of √-1 (Paul J. Nahin; 1998)
12. Dr. Euler’s Fabulous Formula (Paul J. Nahin; 2006)
13. Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers (Dan Rockmore, 2006)
14. Infinity and the Mind (Rudy Rucker, 2004)
15. The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics (Karl Sabbagh, 2004)
16. The Prime Numbers and Their Distribution. (Tenenbaum and Mendès, 2000)
http://www.jorgenveisdal.com/files/jorgenveisdal-thesis13.pd...
So, it's more the language of math that makes it hard to understand than it is about the math itself.
Quantum chaos, random matrix theory, and the Riemann ζ function
http://www.cims.nyu.edu/~bourgade/papers/PoincareSeminar.pdf
What does "larger than zero" mean for complex numbers?
The Wikipedia article on the gamma function
https://en.wikipedia.org/wiki/Gamma_function
states that "The gamma function is defined for all complex numbers except the non-positive integers. For complex numbers with a positive real part, it is defined via a convergent improper integral ..."
So maybe that is what the OP means: the gamma function Γ(z) can be defined by via convergent improper integral for all complex numbers of z with real part greater than zero.
?
Wikipedia then points out that:
"This integral function is extended by analytic continuation to all complex numbers except the non-positive integers (where the function has simple poles), yielding the meromorphic function we call the gamma function."
No, I had no idea what he meant. His statement about complex z larger than 0 came right after he gave a plot of gamma for real z from -6 to 6.
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.
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.
This article is not about the monitoring project.
[1]: http://riemann.io/
followed by a function of one real variable with an infinite sum of expressions containing integer r.
Sorry, but little things like these completely put me off in math articles.
If you don't mind me asking, what is so off-putting about these types of mistakes? I'd like to do more personal and professional technical writing similar to this but I find it a bit daunting, especially when I consider that my audience could know much more about a subject than I do. Do you have any suggestions on how I could or should write articles that may contain errors so I don't upset readers?
Basically when I see something like this I involuntary start thinking: "Is it really a typo? Or is it the writer doesn't understand what he is writing about, because he cannot explain it to me in a clear and precise way?". Maybe this "BS filter" I have is a bad thing for me :(
About writing articles - I think one shouldn't worry about audience knowing more about a subject - it's inevitable, and it's what discussions in comments are for. So, errors due to not knowing something are ok, I think. Other errors can easily be corrected by proof reading (by a friend or by oneself after a day). That what I would do if I actually decided to start a blog :)
The problem for me is that my way of reading math stuff is to assume that whatever is said is being true and fit it into a growing mental model as I read. Many other students I know do that. Once you get into the flow of it this can get pretty fast without losing out on comprehension. When you come across mistakes, it leads to a speed bump, and you have to stop and re-collect your thoughts after this. This only happens to me for math and physics, not other kinds of technical writing, but it happens.
This doesn't turn me off an article, it's just super annoying.
Typos and stuff are fine, it's the "plausibly true" stuff that gets you. I suggest spending more time proofreading but not worrying more about it.
Why are people still posting things on Medium?
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.
end meta
This is a good post, and it should be put somewhere that will keep it.
That said, "RH" stuff now:
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.
The people who think of him as a great thinker as it applies to music are completely out of their minds.
/trolling, sort of. a little.
Rant redacted.