Describe the Riemann zeta function and some of its properties as intuitively as possible.
Try to make the reader extremely interested in the Riemann zeta function.
>>7746199
https://en.wikipedia.org/wiki/On_the_Number_of_Primes_Less_Than_a_Given_Magnitude
>>7746199
keh
As a metaphor it can be said that the Riemann Zeta function is that where you have a mechanical bull and it's rider; the rider plays the function of meta-physical resistance to the bulls tetrating speed, the tetration is caused by the allocation of input compared to the riders proficiency, but there's a catch, the resistance that the rider portrays is created as the input of energy is put up is increased exponentially to the point that the bull bulls energy distribution when put into further energy heights becomes so chaotic that the rider has full operation distribution over the bull and just before the mechanical bull snaps off its axis the rider becomes the operator, and the distribution flips so that the maximum occupancy of the bull increases and the sectoring of the operator that is regulating input energy, although the dial already be at max; that the dial seems to just be sitting at minimum input.
>>7746227
Thanks for the in-depth description, it really shed some light on the subject matter.
>>7746243
You know that's just a bunch of gibberish, right? You're being sarcastic, please?
>>7746227
Fantastic
>>7746227
Thank you so much anon, this really helped
>>7746227
Wow, marvelous insight!
>>7746243
>>7746277
>>7746374
>>7746405
Thanks
Would you like me to do another? Your pick
>>7746487
Tell us very specifically about the relation of the Riemann zeta function and primes.
>>7746487
inter-universal teichmuller theory
>>7746833
this
/sci/, can't you do this?
>>7747084
Bernhard Riemann was a master of fourier analysis, a more applied child of complex analysis, and from there he reached out into different mathematical subjects. For example, Riemann surfaces are two-dimensional real surfaces and give us a good image of what complex valued functions ought to look like. And Riemann geometry now famously provides the tools for general relativity.
About 150 years ago, Riemann must have been bored as he wrote a short paper on number theory, the only number theory paper he wrote in his life. It’s not really a number theory paper in the sense people did number theory before. It’s about complex analysis, and more like a conglomerate of what the fuck moment. „Oh btw. guys, have you tried this?“
The function in (for now) a real variable s > 1 defined by
[math] \zeta(s) := \sum_{n=1}^\infty n^{-s} [/math]
was looked at by Euler and owes it’s relevance to [math] (y · z)^x = y^x · z^x [/math]. The sum is over all numbers and those are multiplicatively build up from primes and you should really read it as
[math] 1 + 2^{-s} + 3^{-s} + 2^{-s} \, 2^{-s} + 5^{-s} + 2^{-s} \, 3^{-s} + 7^{-s} + 2^{-s} \, 2^{-s} \, 2^{-s} + 3^{-s} \, 3^{-s} + \dots[/math]
The Euler-Product is a compact product representation of that sum,
[math]\prod_{p} \frac {1} {1-p^{-s}} = \frac {1} { (1-2^{-s}) (1-3^{-s}) (1-5^{-s}) (1-7^{-s})\dots} [/math]
It’s a product over all primes.
By the rules of the logarithm, a log of a product is a sum of logs, and that’s the starting point for Riemann.
He first considers the function in s for complex numbers - this works for all s except at s=1. What comes next is the transition of something inherently discrete to his own playground of function theory. And it’s a little mad.
The object of interest is a function that counts primes up to some real number x.
(cont.)
Let
[math] \Pi(x) = \sum_{p^n < x} \frac {1} {n} [/math]
be the function that sums those [math] \frac {1} {n} [/math] for which, for all primes p, the expression p^n is still below x.
E.g.
[math] \Pi(20) = {(1 + \frac {1} {2} + \frac {1} {3} + \frac {1} {4} )}_{2^1, 2^2, 2^3, 2^4 < 20} + {(1 + \frac {1} {2} )}_{3^1, 3^2 < 20} + {(1)}_{5^1 < 20} + {(1)}_{7^1 < 20} + {(1)}_{11^1 < 20} + {(1)}_{13^1 < 20} + {(1)}_{17^1 < 20} + {(1)}_{19^1 < 20} = \frac {115} {12} [/math]
Three points:
-The expansion of the log is with coefficients (-1)/n.
-We have [math] - s \frac {d} {dx} \frac {1} {x^s} = x^{-s-1} [/math] and so we
[math] (p^{-s})^n = (p^n)^{-s} = s \int_{p^n}^\infty x^{-s-1} \, dx [/math]
-Also, note that if H(y-3) is a step function with the jump to 1 at y=3, then
[math] \int_0^\infty f(y)\, H(y-3) \, dy = \int_3^\infty f(y) \, dy [/math]
Now the man does this:
[math] \log \zeta(s) = \sum_{p} \sum_{n=1}^\infty \frac { p^{-ns} } {n} = s \sum_{p} \sum_{n=1}^\infty \frac {1} {n} \int_{p^n}^\infty x^{-s-1}\, dx = s \int_0^\infty x^{-s-1} \Pi(x) dx [/math]
What is the next step of his master plan?
He’s like: Did you guys know that this is a Fourier transform (Mellin transform after x to exp(-s)) which you can invert?
[math] \Pi(x) = \frac {1} {2 \pi i} \int_{c-i\infty}^{c+i\infty} x^s \, \frac {1} {s} \, \log \zeta(s) \, ds [/math]
Suddenly the values of your discrete function are determined by the poles of a complex function.
What the fuck?
And btw., indeed, where [math]\zeta(s)[/math] is zero, the log of it explodes.
There are some areas in solid state physics where we have exactly this situation, btw. The poles of the partition function over all possible states (the ones that define the entropy of the system) determine properties such as transition from gas to liquid to solid (think ice), and so on.
(cont.)
Also, the way of taking the sum to the complex plane looks somewhat like this:
First he observes that a integration variable substitution x -> n·x in the definition of the Gamma function
[math] \Gamma(s) := \int_0^\infty x^{s-1} e^{-x}\,dx [/math]
let's you write
[math] n^{-s} = \frac {1} {\Gamma(s)} \int_0^\infty x^{s-1} e^{-nx}\,dx [/math]
and then, recognizing the geometric series
[math] \sum_{n=1}^\infty (e^{-x})^n = \frac{1} { e^{-x}-1} = \frac{1} {x} - \frac{1}{2} + \frac{1}{12}x+O(x^2) [/math]
([math] \frac{1}{12} [/math], see >>7746983)
give you
[math] \sum_{n=1}^\infty n^{-s} = \frac{1} { \Gamma(s) } \int_0^\infty \frac{x^{s-1}} { e^x-1} \, {\mathrm d}x [/math]
He takes the integral into the complex plane, where he [math] \frac{1} { e^x-1} [/math] diverges periodically in steps of [math] 2\pi\,i [/math].
Playing around with the exponential function and doing some mirroring when you have symmetris, he discovers that the function obeys a reflection formula
[math] \zeta(s) = (2\, \pi)^s\,(\sin (\frac {\pi s} {2} )/\pi)\,\Gamma(1-s)\ \zeta(1-s) [/math]
and now for negative s you know the value once you know it for positive ones. E.g.
[math] \zeta(-1) = (2\, \pi)^{-1}\,(\sin (\frac {\pi (-1)} {2} )/\pi)\,\Gamma(1-(-1)) \,\zeta(1-(-1)) [/math]
[math] = \frac {1} {2} \frac {1} {\pi^2} \, (-1) \, 1! \, \sum_{n=1}^\infty \frac {1} {n^2} [/math]
[math] = -\frac {1} {2} \frac {1} {\pi^2} \frac {\pi^2} {6} [/math]
[math] = -\frac {1} {12} [/math]
>>7747190
I literally just summarized Riemanns paper:
https://upload.wikimedia.org/wikipedia/commons/c/cb/Ueber_die_Anzahl_der_Primzahlen_unter_einer_gegebenen_Gr%C3%B6sse.pdf
Translation here:
http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf
>>7746830
Well that can be remedied as simply as what binary distribution language you're using to regulate the input into axis definition of the bull, math doesn't catch the rhythm to well because programming compared rhythms compared to math rhythms are relatable as the integral function in comparison to the seeds in a banana as a programming language is to finding all the patterns in a spiral that is shaped like a spring being looked at from a diagonal value; where the seeds in the banana add to the compression of the formula of the program and the integral is that of the compression value in the spring making for the the of values to require more input from the program and distributing the constance of the banana seeds to the point of there being an 'ultimate' prime where one could say there is a seed is delegated to each end of the banana, this could be said to be infinity in non math intuitivism; and in the case of the integral in regards to the density of the spring is the other end of the kaleidoscope, so to speak, and at this there has to be electromagnetism acknowledged where the, as I like to see it, tetrahedron chain contriving as a more obscure definition of what exposes electromagnetism, that the integral function on the spring acts with this system and defines the principles of a less gun shy rhythm of an inverse integral fractal in a sense, or comparable to the mandalbrot set's construction if you were to go from the peak of it to our current rendition of c which is also how one could derived most of what we know of relativity; but I digress, those two analogies as they compliment eachother are how the actions of primes derive themselves, as quite a mess I might add... kind of similar to a tooth edged coin rolling and getting ever larger while the teeth maintain a rhythm of every 1/(nth+inf) even being added to every odd, in the Riemann zeta function
If you are adverse to this description, provide a list of aspects and I'll have a shot at it
>>7746833
You see that tattoo on that guys shoulder of the triangle/cricle?
Just keep expanding the potential dimensions on that and you've get Teichmuller Theory, line all of them up regardless of where their positions are and you've got Inter-Universal, keep in mind when ever you're doing physics you should translate each variable into |String->Ball->Wave|->Ball->String->Ball->Wave->etc. with variations of order of them and derivation the base equations making their structures more equipable to the rest of the factoring
>M-Theory fan here :^)
>>7747330
in regards to the last line of that description there the n'th being regarded as even not the equation :]
>>7747339
Have you met the good doctor?
Your theories remind me of him.
>>7747980
Probably out my generations quasi pop-sci awareness, so no, not likely
>tell me more \:^] XD
>>7748010
Went by the name of SSDoctor. He supported theory on branes and stated that we're all brilliant liquid diamond quantum computers that operate on micro black holes (electrons?)
>>7748010
You even used one of his trademark pictures, the one with the firefly wings.
Are you the good doctor?
>>7746266
>unable to pick up on sarcasm
Are you actually autistic?
>>7748783
> micro black holes (electrons?)
All fields have the same principles, how they dialate their enviroment is subject to what they're composed of
Have you never really found it disturbing that a photon can be absorbed by anything at all and not transfer the whole potential of its internal regulation?
>>7749304
Well I came across Astro boy once in a dream
>.>"
<_<"
/:^)
