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A Friendly Introduction to The Riemann Hypothesis [pdf] (jhu.edu)
128 points by signa11 on July 12, 2016 | hide | past | web | favorite | 23 comments



Most people have heard of the Sieve of Erathosthenes: put all positive integers in a list, and delete the number one. Now mark two as prime and delete all multiples of two. Now mark the next remaining number prime and delete all multiples of that number. Proceed ad infinitum.

There's a randomised counterpart to the Sieve of Erathosthenes called the Hawkins ransom sieve: Delete the number 1. Mark two as prime, and delete all the larger integers with probability 1/2. Now mark the smallest remaining integer (n, say) prime, and delete all the remaining larger integers with probability 1/n. Proceed ad infinitum.

One of my favourite results in mathematics is that the Riemann hypothesis (i.e. the error bound for the prime number theorem) holds for the Hawkins random primes with probability 1. The proof is only about five pages: https://eudml.org/doc/89234


That's very cool; thanks for sharing.


Interesting, never heard of a non-deterministic Sieve of Eratosthenes before.


An interesting read! Unfortunately the author omits the mindblowing (at least for me) proof of why the series

Z(s) = 1 + 1/(2^s) + 1/(3^s) + 1/(4^s) + ...

is equal to the infinite product over the terms "1/(1-p^(-s))", with one term for each prime number p.

This proof does not need a lot of math [1], and if you know the formula for a geometric series, you can also "prove it" by hand-waving: Each term in the product is a geometric series, and if you multiply all these series out, the result is a sum over all possible permutations of prime-factored numbers (well, the inverses of them). Since each natural number has exactly one such represnetation, it's effectively a sum over all 1/(n^s).

Of course, you need some rigor to show that you are actually allowed to rearrange terms of an infinite product of inite sums like that...

[1]: https://en.wikipedia.org/wiki/Proof_of_the_Euler_product_for...


Long before I went to university, I became obsessed with the Riemann hypothesis and one of the first things I put on my site was a simple explanation of the Riemann Hypothesis. This was before I ever actually studied math formally, so it's not a very authoritative explanation. For whatever reason, this explanation is now on the first page of search results in google.ca for "Riemann hypothesis" in Canada, and at the top of page 2 on google.com. I get lots of random blog posts citing the explanation I wrote, and every once in a while I get emails from crazy people who think they have solved it providing me with their non-sensical proofs.

The problem itself is amazing (even though I don't really understand it). The infinite series is so simple, but it generates so much interesting complexity when you look at it visually:

https://upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Ze...

http://mathworld.wolfram.com/RiemannPrimeCountingFunction.ht...


The Riemann hypothesis is equivalent to a number of interesting statements. One that I like is this.

The Möbius function, defined by http://mathworld.wolfram.com/MoebiusFunction.html, shows up in various sieving operation. It is 0 for a random integer with probability (1 - 6/pi^2) and otherwise has apparently even odds of being +-1.

Its sum is called Mertens function, see http://mathworld.wolfram.com/MertensFunction.html. The Riemann conjecture is equivalent to saying that the growth of Mertens function is o(n^(0.5+e)) for every e > 0. For a long time it was thought to be bounded above by n^0.5, but this is known to be wrong.

If you replace the Möbius function with a function that is -1, 0 and 1 with the right probabilities, then with probability 1 it is O((n * log(log(n))^0.5). Which would prove the Riemann hypothesis. (This intuition would have kept Mertens from conjecturing that it is bounded by +-sqrt(n)...)


If you are interested in a longer account, I highly recommend mathematician/popularizer Marcus DuSautoy's Music of the Primes. https://en.m.wikipedia.org/wiki/The_Music_of_the_Primes


I would add John Derbyshire's 'Prime Obsession' to that, which adds a nice historical context to the issue.


This is fantastic! I know next to nothing about number theory (despite my interest in primes) and this had me in stitches. Thanks for the link.


That article was really a great refresher on something I barely remember from undergraduate courses. I'm left a little unfazed about the style of prose, but I guess that's the price you have to pay for dat dankness, yo.


Yes, I liked the content in spite of the "trolling," since there is enough of actual math, the author actually knows the field and it is presented simple enough to make the math path enjoyable.

I'm sure it would work good even with less or no "trolling." It is possible to make the approachable text even without the jokes based on misinformation. I see the author made the book with that in the title, so it was obviously a selling point for him, but the quite clear approach to the math material (when he reader manages to ignore the trolling) is actually good. I have nothing against the funny presentation of the actual biographical facts however.

The misinformation is, however, problematic. The author even states in the foreword that he "hopes that some of those will appear in Wikipedia." Unfortunately, it really can happen.


I'm enjoying this text immensely. Math should be presented this way more often.


A new pedagogical paradigm?


I wouldn't necessarily go that far. My personal taste may not be representative.


Every now and then someone posts some humorous easily-digestible technical content to Hacker News and commenters say stuff like "All mathematics/science/software development textbooks should be written this way", and some commenters throw out a few similar links, and the story slowly drops down the HN front page, as stories do, and then it's mostly forgotten, until the next one comes along.

If education matters, and if human intelligence augmentation matters, then we should be researching how to make difficult technical material more easily digestible, for example by enhancing it with an appropriate level of humour.

Millions of dollars are being spent on making machines more intelligent (or is it billions?). Someone should be spending some time, money and effort on making people smarter.

Also, compared to other possible human intelligence augmentation technologies, writing textbooks in a more humorous style is less intrusive than taking "smart pills" or placing electrodes inside your brain.


Through the history of mankind we've been researching ways to make difficult technical material more easily digestible. And may the invention of print itself, and the subsequent invention of "the textbook" be a showcase example of this.

Though humorous style is less intrusive than taking "smart pills" or placing electrodes inside our brains, it is very context-sensitive. I believe humour should definitely be used more often in oral presentations of complicated material, where the presenter is able to "feel" the context of those listening; but maybe not so much in written text.

To make a point, there are many technical textbooks that use funny examples and exercises as a way to draw students attention - and that does generally work quite well - but it can be quite a tiring read if you are going through the material multiple times.


If you're looking for something with a little more detail (but that still doesn't require a lot of mathematical background), you should check out https://github.com/williamstein/rh


The author has a book, Trolling Euclid[0], on Amazon.

[0]https://www.amazon.com/Trolling-Euclid-Irreverent-Mathematic...


This style of humor reminded me very strongly of Dave Barry.

https://en.wikipedia.org/wiki/Dave_Barry


james-mickens has a _very_ similar style as well. for example, if you have not read 'the night watch' (http://scholar.harvard.edu/files/mickens/files/thenightwatch...) check it out.

you might like it :)


That is definitely similar! Thanks for the suggestion.

I absolutely adored Dave Barry's work when I was in middle school (I often couldn't read a single paragraph aloud without laughing out loud) but I find it less enjoyable today. Maybe my taste in humor has shifted somewhat.


Someone needs to rewrite Rudin in this way.


Can I double upvote!




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