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The Riemann Hypothesis

golem.ph.utexas.edu

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Re: The Riemann Hypothesis

#21

Post by j9461701 »

Cybiote wrote: post #13

Here's a quite friendly elucidation by the inimitable Avi Wigderson, from: https://www.ias.edu/ideas/2009/wigderson-randomness-pseudora... > 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 > 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 . . . > Such a sequence can be thought of a…

>monomer-dimer problem

Oh hey, I did my undergrad thesis on that! It generates neat looking graphics:

https://imgur.com/a/Z6hySAw

Re: The Riemann Hypothesis

#22

Post by jacobolus »

ncmncm wrote: post #14
liquidise wrote: post #10

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. Most fascinating to me is that many theories are effectively 1-way functions. Entire branches of mathematics have been developed to prove otherwi…

The notion that multiplication is ("just") repeated addition is the hardest misconception to overcome blocking progress in mathematics. Repeated addition is an algorithm that works when you have a non-negative integer argument. Imagining that the algorithm defines the operation limits conception.

> hardest misconception to overcome

This seems like an exaggeration.

Re: The Riemann Hypothesis

#23

Post by why-el »

liquidise wrote: post #10

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. Most fascinating to me is that many theories are effectively 1-way functions. Entire branches of mathematics have been developed to prove otherwi…

Well, I am not sure if you've come across some of Bertrand Russell's work, but he qualified Mathematics as having what he called "supreme beauty". The older I get, and I am not a mathematician by any stretch of the imagination, the more I get what he means. This stuff is surely a gift for all of us.

Re: The Riemann Hypothesis

#24

Post by oneepic »

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... :) )

The article is spot on. I've had so many moments where the math looks so fishy that it seems like R has 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.

Re: The Riemann Hypothesis

#26

Post by schwurb »

Yajirobe wrote: post #19

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.

It's the same story as with many other deep and important math problems: Practically, not much would change. Many papers already assume RH - we would just have confirmation that they are reality then. What would change is that we would likely have novel and powerful methods that were used in the proof. This is what most of the buzz is about. For example, the proof of Fermats Last Theorem introduced novel connections between sub fields of mathematics.

Re: The Riemann Hypothesis

#27

Post by schwurb »

ddxxdd wrote: post #20

1. I'm a big fan of John Baez. 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.

> P=NP in that a solution can be used to attack RSA encryption.

Note that 1) P=NP does not necessarily give raise to any polynomial algorithm that solves a NP problem. The proof would prove the existence of one such algorithm, but it might well never be found (which is the current status quo) 2) even if it would be polynomial, it could still run longer than the heat of the universe. O(n) = n^10000000 would still be a polynomial runtime for example. The second reason is why Donald Knuth does think that P=NP might be possible.

Re: The Riemann Hypothesis

#28

Post by sls »

apalmer wrote: post #17
ncmncm wrote: post #14

Earlier quoted context omitted.

The notion that multiplication is ("just") repeated addition is the hardest misconception to overcome blocking progress in mathematics. Repeated addition is an algorithm that works when you have a non-negative integer argument. Imagining that the algorithm defines the operation limits conception.

I would like to understand what multiplication is in your (and i guess advanced mathematics)

The comment to which you replied says that thinking about multiplication as "just" repeated addition is problematic, so let's look at that.

Consider 3 x 2. If we take that approach, it seems ok - we understand it to mean "add together 3 2's" - 2 + 2 + 2, which gives the correct answer of 6.

What about -3 * -2? What does it mean to add a negative number of times?

What about pi * pi? What does it mean to add something pi times?

What about the matrix M * the matrix N?

etc. The parent's point is that this repeated addition thing is just an algorithm one can use to calculate a multiplication for some operands, specifically whole numbers, not a general definition of the operation of multiplication.

Re: The Riemann Hypothesis

#29

Post by monktastic1 »

oneepic wrote: post #24

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... :) ) The article is spot on. I've had so many moments where the math looks so fishy that it seems like R has to equal 1/2 (ie h…

> Wish I could share my leads, but I kinda want the money and glory... :)

Well, like Prof Baez says, unless you've solved other major open problems before, it's probably unwise to believe you'll be the one to crack it. You are likely to gain more by sharing your leads and seeing what mathematicians have to say.

Re: The Riemann Hypothesis

#30

Post by jamiek88 »

j9461701 wrote: post #21
Cybiote wrote: post #13

Here's a quite friendly elucidation by the inimitable Avi Wigderson, from: https://www.ias.edu/ideas/2009/wigderson-randomness-pseudora... > 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 > 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 . . . > Such a sequence can be thought of a…

>monomer-dimer problem Oh hey, I did my undergrad thesis on that! It generates neat looking graphics: https://imgur.com/a/Z6hySAw

Looks like an X-ray pic of a system on a chip.
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