Here's a quite friendly elucidation by the inimitable Avi Wigderson, from: <a href="https://www.ias.edu/ideas/2009/wigderson-randomness-pseudorandomness" rel="nofollow">https://www.ias.edu/ideas/2009/wigderson-randomness-pseudora...</a><p>> Let’s elaborate now on the connection (explained on the cover of this issue) of the Riemann Hypothesis to pseudorandomness. Consider long sequences of the letters L, R, S, such as<p>> S S R S L L L L L S L R R L S R R R R R S L S L S L L . . .<p>> Such a sequence can be thought of as a set of instructions (L for Left, R for Right, S for Stay) for a person or robot walking in a straight line. Each time the next instruction moves it one unit of length Left or Right or makes it Stay. If such a sequence is chosen at random (this is sometimes called a random walk or a drunkard’s walk), then the moving object would stay relatively close to the origin with high probability: if the sequence was of n steps, almost surely its distance from the starting point would be close to √n. For the Riemann Hypothesis, the explicit sequence of instructions called the Möbius function is determined as follows for each step t. If t is divisible by any prime more than once then the instruction is Stay (e.g., t=18, which is divisible by 32). Otherwise, if t is divisible by an even number of distinct primes, then the instruction is Right, and if by an odd number of distinct primes, the instruction is Left (e.g., for t=21=3x7 it is Right, and for t=30=2x3x5 it is Left). This explicit sequence of instructions, which is determined by the prime numbers, causes a robot to look drunk, if and only if the Riemann Hypothesis is true!
Mathematics is a uniquely beautiful field to me. The commutative property has always struck me as special in its own way. 2 x 3 = 3 x 2 feels so obvious, but multiplication is really just addition, and 2 + 2 + 2 = 3 + 3 is far less intuitive, yet states the very same claim.<p>Most fascinating to me is that many theories are effectively 1-way functions. Entire branches of mathematics have been developed to prove otherwise trivially stated claims. It is something to marvel at, if from a safe distance.
3Blue1Brow has a nice video on it <a href="https://www.youtube.com/watch?v=sD0NjbwqlYw" rel="nofollow">https://www.youtube.com/watch?v=sD0NjbwqlYw</a>
I really wish the Riemann-Zeta Function were more often explained in terms of a prime number sieve. It's actually not particularly difficult to follow and the connection between the function and the distribution of primes would be completely obvious.
If you've got enough math background to follow it, Harold Edwards' <i>Riemann's Zeta Function</i> is a gem of a book and available inexpensively from Dover. There is an English translation of Riemann's paper at the end of the book. I spent a worthwhile few weeks of spare time working my way through the paper ( and a lot of pencil & paper to work "between" the steps in the paper -- math is not a spectator sport ) with a longish diversion diving into the gamma function along the way.
I think "don't try to prove the Riemann hypothesis" is only part of the iceberg that includes "you may want to prefer theory building over problem solving" and "we're not getting any medals here". It's interesting that this was written by one of the category theoretic schools; the category theorists that I studied under are quite wary of things like the RH. After all, Saunders Mac Lane never won a Field's medal. I am not throwing shade, but it's exceedingly difficult to try to judge (any) mathematician's "worth" in the way a prize or medal does in popular media.
Every now and then I try to delve into the frightening world of math. Then I see something like this, and start to feel very tired.<p>Then I think, “My hair is already falling out. Do I need something like this to accelerate the process?”
If this interests you, I would strongly recommend reading <a href="http://www.riemannhypothesis.info/2014/10/tossing-the-prime-coin/" rel="nofollow">http://www.riemannhypothesis.info/2014/10/tossing-the-prime-...</a> this explains the relation between random walks and the RH in a surprisingly easy to understand fashion.
My absolute favorite numberphile video is how pi occurs in a peculiar way with Riemann Zeta/Mandelbrot. Maybe it only amazes me because I don’t have a PhD in math, but it just seemed so strange how pi shows up in this video.<p><a href="https://youtu.be/d0vY0CKYhPY" rel="nofollow">https://youtu.be/d0vY0CKYhPY</a>
Why does everybody think that by virtue of math ought to be nice, such a nice hypothesis ought to be true? Isn't it just a form of the survivorship bias that we observe only nice side of math? What if this hypothesis stands true for all N < 10^10^10^467+17, and then suddenly it doesn't? Perhaps to make a breakthru in math (and physics) we need to consider the possibility that the reality can be ugly and counterintuitive and beyond a certain complexity level, math and physics cannot be described by nice formulas.
So if RH is proven, what actually changes? As far as I know, there are tons of theorems that already presuppose RH to be true There wouldn't suddenly be an insight into how to find larger primes, for example.
1. I'm a big fan of John Baez.<p>2. I'm getting the impression from this article that solving the Riemann Hypothesis is similar to solving P=NP in that a solution can be used to attack RSA encryption.
I've been making a serious attempt at solving it but I'm not a mathematician. Even still I have a few good leads yet to pursue, and I learned a ton about the practice of mathematics that I never would've learned otherwise. (Wish I could share my leads, but I kinda want the money and glory... :) )<p>The article is spot on. I've had so many moments where the math looks so fishy that it seems like R <i>has</i> to equal 1/2 (ie hypothesis is true), but I just don't have the facts to prove it. In particular, it's really hard to evaluate the infinite sums you find working thru the problem. I actually believe that there's a good chance the hypothesis is false but we'll see someday.