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The Riemann Hypothesis

golem.ph.utexas.edu

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Re: The Riemann Hypothesis

#51

Post by chii »

emilgouliev wrote: post #46
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…

2 + 2 + 2 = 3 + 3 becomes much more obvious when you think of a 3x2 rectangle and rotate it by 90 degrees to a 2x3 rectangle.

but then isn't this just a tautological argument? why does multiplying the sides of a rectangle give you the area?

Re: The Riemann Hypothesis

#52

Post by dboreham »

chii wrote: post #51
emilgouliev wrote: post #46

Earlier quoted context omitted.

2 + 2 + 2 = 3 + 3 becomes much more obvious when you think of a 3x2 rectangle and rotate it by 90 degrees to a 2x3 rectangle.

but then isn't this just a tautological argument? why does multiplying the sides of a rectangle give you the area?

Because of little squares being added.

Re: The Riemann Hypothesis

#54

Post by sls »

jacobolus wrote: post #33
sls wrote: post #28

Earlier quoted context omitted.

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…

If we want to define π × π , the best we can do is some kind of algorithm for generating an approximation to π (e.g. as a continued fraction or as a positional decimal fraction) to any desired degree, and then an algorithm for multiplying such approximations. We can prove some bounds on the error introduced by our multiplication algorithm, and that gives us a way of approximating the product to any desired precision.…

I don't agree that a definition of multiplication that includes transcendentals such as pi must be numerical. In fact, to the extent that numerical approximations are approximations _of_ something, the thing they would be approximating is the actual value of the operation of multiplication being applied to pi and pi. The only reason we're talking about algorithms at all was to distinguish between them and definitions.

Re: The Riemann Hypothesis

#55

Post by teh_infallible »

arkano wrote: post #11
teh_infallible wrote: post #2

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. Then I think, “My hair is already falling out. Do I need something like this to accelerate the process?”

As a mathematician, I have noticed that people that like to build something that, for example, can fly, start from a paper plane; they don't get discouraged because they can't yet build a 737 aircraft. However, in math, you need a lot of experience before you can even judge whether a problem is in fact a 737 and not a paper plane (and even then, you can be mistaken). I often see students discouraged because of this a…

Thank you for your elegant reply. I really appreciated it because I am a non-engineer with a 3d printer, so your paper plane example was quite relevant.

“Tinkering” with mathematical concepts is fundamentally different from the kind of tinkering I can do, but looking from the outside in.. I’ll probably pick up a calculus textbook one day :)

Re: The Riemann Hypothesis

#56

Post by jacobolus »

sls wrote: post #54
jacobolus wrote: post #33

Earlier quoted context omitted.

If we want to define π × π , the best we can do is some kind of algorithm for generating an approximation to π (e.g. as a continued fraction or as a positional decimal fraction) to any desired degree, and then an algorithm for multiplying such approximations. We can prove some bounds on the error introduced by our multiplication algorithm, and that gives us a way of approximating the product to any desired precision.…

I don't agree that a definition of multiplication that includes transcendentals such as pi must be numerical. In fact, to the extent that numerical approximations are approximations _of_ something, the thing they would be approximating is the actual value of the operation of multiplication being applied to pi and pi. The only reason we're talking about algorithms at all was to distinguish between them and definitions…

Well, that’s a philosophical rather than mathematical question. I don’t really believe in a concept of “actual value” outside of the context of computations (though I don’t mind conceding it as a matter of convenience and social convention, since the distinction almost never matters for practical purposes). I am not an expert, but mathematicians have investigated this, https://en.wikipedia.org/wiki/Computable_analysis

We can treat π purely symbolically if we like, but as soon as we want to do anything useful with it we need some kind of approximation or algorithm.

Re: The Riemann Hypothesis

#57

Post by drdeca »

schwurb wrote: post #27
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…

Isn’t there some (highly impractical) algorithm which dovetails through different Turing machines, in a way that has an asymptotically optimal runtime for a given problem, just with really terrible constants?

I thought we knew an algorithm that, if P=NP, would solve NP problems in P time, (but with absurd constants), and otherwise solves the problems is worse than polytime.

But I could be remembering this totally wrong.

Re: The Riemann Hypothesis

#58

Post by unixhero »

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

Something's going in there.

Re: The Riemann Hypothesis

#59

Post by mikorym »

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.
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