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The Riemann Hypothesis

golem.ph.utexas.edu

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Re: The Riemann Hypothesis

#61

Post by teekert »

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

I did a Masters in Biophysics, the topic was diffusion of proteins in cell membranes, they usually show random walks (albeit restricted to compartments or showing distinct speeds (bound/unbound?)). That graphs does not look like a random walk, so the Riemann Hypothesis is false?

Re: The Riemann Hypothesis

#62

Post by teekert »

chii wrote: post #50
wodenokoto wrote: post #44

Earlier quoted context omitted.

> (e.g., t=18, which is divisible by 32) Did they mean 3x3x2 instead of 32? There is no way 18 is divisible by 32 in any common sense of divisible. > 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! So, do they look drunk for large walks? That sounds like something that is easily computed for tens of thousand…

what does "look drunk" actually mean tho? it's a bit of a weird property...

The drunken man is moving away from where he started (any point in time can be labelled as "start") at a speed of about the square root of his linear speed (speed from his point of view). And the direction is random. This can also be 1D motion. Actually in 1 and 2D it is likely that the drunken man hits his starting point again at some point, it goes to 0 fast in 3D.

Re: The Riemann Hypothesis

#63

Post by kmm »

wodenokoto wrote: post #44
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…

> (e.g., t=18, which is divisible by 32) Did they mean 3x3x2 instead of 32? There is no way 18 is divisible by 32 in any common sense of divisible. > 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! So, do they look drunk for large walks? That sounds like something that is easily computed for tens of thousand…

It has been computed for at least 10^16 steps, and it sure does look drunk, but sadly that does not constitute a proof.

https://en.m.wikipedia.org/wiki/Mertens_function

Re: The Riemann Hypothesis

#64

Post by red75prime »

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?

Multiplication in the form we know it arose from practical needs of area measurement, not as some abstract operation in semirings and rings.

Multiplying the sides gives you the area because multiplication was invented/discovered to do that.

Re: The Riemann Hypothesis

#65

Post by teekert »

Double_Cast wrote: post #47
apalmer wrote: post #17

Earlier quoted context omitted.

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

The problem is that there's different types of multiplication. The fact that multiplication over the Reals can be thought of as "iterated addition" is a consequence of commutativity. But mathematicians often generalize multiplication to various contexts as an arbitrarily-defined transformation, which may or may not be commutative. E.g. multiplication over the Reals is commutative; multiplication over Complex Numbers…

It doesn't, but isn't that, in this case, only because you math gals and guys simply define x^0 to be one? There is no logical reason from a geometry standpoint indeed but you needed a definition.

Re: The Riemann Hypothesis

#67

Post by wmp56 »

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.

Re: The Riemann Hypothesis

#68

Post by ukj »

Double_Cast wrote: post #47
apalmer wrote: post #17

Earlier quoted context omitted.

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

The problem is that there's different types of multiplication. The fact that multiplication over the Reals can be thought of as "iterated addition" is a consequence of commutativity. But mathematicians often generalize multiplication to various contexts as an arbitrarily-defined transformation, which may or may not be commutative. E.g. multiplication over the Reals is commutative; multiplication over Complex Numbers…

On-point.

Exponentiation is not like multiplication because it's not associative.

(2^3)^4 != 2^(3^4)

Re: The Riemann Hypothesis

#69

Post by wodenokoto »

chii wrote: post #50
wodenokoto wrote: post #44

Earlier quoted context omitted.

> (e.g., t=18, which is divisible by 32) Did they mean 3x3x2 instead of 32? There is no way 18 is divisible by 32 in any common sense of divisible. > 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! So, do they look drunk for large walks? That sounds like something that is easily computed for tens of thousand…

what does "look drunk" actually mean tho? it's a bit of a weird property...

As the parent poster quotes, drunk looks like:

> if the sequence was of n steps, almost surely its distance from the starting point would be close to √n.

Of course that raises the question "What is close?"

Re: The Riemann Hypothesis

#70

Post by Double_Cast »

teekert wrote: post #65
Double_Cast wrote: post #47

Earlier quoted context omitted.

The problem is that there's different types of multiplication. The fact that multiplication over the Reals can be thought of as "iterated addition" is a consequence of commutativity. But mathematicians often generalize multiplication to various contexts as an arbitrarily-defined transformation, which may or may not be commutative. E.g. multiplication over the Reals is commutative; multiplication over Complex Numbers…

It doesn't, but isn't that, in this case, only because you math gals and guys simply define x^0 to be one? There is no logical reason from a geometry standpoint indeed but you needed a definition.

(x^0) = (1) makes sense because lim{y -> 0} (x^y) = (1).

E.g. (10^0.001) = (1.00230524...)

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