Riemann Hypothesis visualised

[@p_square]

one of the most beautiful functions in mathematics

[@kodtech]

is just a Polylogarithm function case....

[@frankansari3457]

One of my favorite KZbin videos ever.

[@Alexander-oh8ry]

Wow, my condolences for your uninteresting life

[@DeanNiemi-r1m]

It is no coincidence we use t for both the imaginary part of the complex argument and a time variable. Reimann is inviting us to walk along the real 1/2 line to inspect the complex codomain values. Think of time as a cursor. Generally, nobody uses the variable t without it meaning time. This part is more advanced, but it can get confusing if later on we wish to use t as the codomain imaginary variable. Then a paradigm shift is in order.

[@nunnyu]

zip up riemanns pants when you are done bro

[@Jon-mi8by]

its almost like gravity of the function is changing

[@frankansari3457]

Would be nice to have t to be also shown (perhaps as growing bar seperately).

[@MarkusShepherd]

Frank Ansari yes, I agree. This would pretty much be the same as plotting the argument and the image it maps to at the same time.

[@JwalinBhatt]

This is awesome, but I always wonder why do we approximate the prime counting function to begin with? Cant we have Riemann like approximation for a smooth curve which passes through the primes: 2,3,5,7,... but not in staircase fashion?

[@lih3391]

Likely its difficult to come up with a function like that, it might just come less naturally from the math

[@JwalinBhatt]

@@lih3391 Could be, but there has to alteast be some attempt or perhaps a formal definition of the problem. Something like a Bohr-Mollerup theorem for primes. People could just try brute forcing or some searching algorithm (Eureqa software) in the space of functions to have some good guesses. There has to be atleast something in this direction, I wonder why I cant find anything.

[@thatkindcoder7510]

Engineers after checking the first 10 zeros to a 2 decimal place precision: "Seems true"

[@mohamedabdelkareem9443]

xD

[@GabriTell]

xdd

[@willyou2199]

at the start z(0).. did you mean z(1/2+0i) ? because z(1/2)= -1.46......, and z(0)= -1/2

[@MarkusShepherd]

Yes, you are right, my mistake. :-( Unfortunately, youtube doesn't let me edit those any more, so we'll be stuck with that error...

[@GorjeCeleb]

hello

[@rainbowbloom575]

How are the zeroes of the function related to prime numbers?

[@MarkusShepherd]

Cristina López well, that's a big question... I answered some of it in my blog, a good article to start would be this one: www.riemannhypothesis.info/2014/10/tossing-the-prime-coin/

[@rainbowbloom575]

@@MarkusShepherd Many thanks xD, I will try to understand

[@rainbowbloom575]

Also, what music plays in the background?

[@MarkusShepherd]

@@rainbowbloom575 I actually don't know. It is one of KZbin's free music that I just chose from their list, but by now so much has changed in their interface that I cannot find the name anymore...

[@someguy4003]

Beautiful

[@DivineRedwood]

*True or false - every time the graph "hits" the origin (0,0) a "prime number" is shown to exist.*

[@denysvlasenko1865]

Incorrect...

[@omerd602]

False - there's much more nuance to it than that, although the values of the zeros do help in determining the locations of primes

[@RSLT]

Beautiful!

[@wisecase2136]

What series do you use to calculate this?

[@MarkusShepherd]

wisecase 2 www.riemannhypothesis.info/2016/04/visualising-the-riemann-hypothesis/

[@minn2410]

A Bow. Told U so..

[@rayubinger9780]

Seen these loops before but nobody explains how they come from the RZ function.

[@MarkusShepherd]

Does this article help? www.riemannhypothesis.info/2016/04/visualising-the-riemann-hypothesis/

[@rayubinger9780]

@@MarkusShepherd Yes, thanks! "These are the values of \zeta(s)ζ(s) as ss goes up the critical line s=\frac12+tis=21​+ti. We start at1 t=0t=0 at the beginning of the video and go all the way up to t=200t=200. \zeta(1/2)\approx-1.4603545\ldotsζ(1/2)≈−1.4603545…, so this is where the values start." Perfectly clear now! But now, unsure why bother? We already know the critical line has an infinitude of zeroes.

[@mechwarreir2]

is there any symmetry to this pattern or is it completely random? I would think its random since the zeta function is composed of an infinite product of primes.

[@MarkusShepherd]

The zeta function itself is highly symmetrical - it has two axes of symmetry, Re z = 1/2 and the x-axis. This means you could have the mirrored spiral going "south" along the critical line. Otherwise we're still trying to understand the pattern of the zeros and why they all line up on Re z = 1/2... ;-)

[@mechwarreir2]

I meant symmetries in polar form. Also I don't think Re(z) = 1/2 is a symmetric axis since there are countably infinite zeros on the analytical continuation to Re(z) < 0 but no zeros right of Re(z) =1/2. As for polar form, I've never seen a more chaotic analytical spiral. Being a hypertrancendental function, you cannot express the zeta function as a solution to a differential equation, otherwise you should be able to use a fourier transform to derive all symmetries and solve the Riemann hypothesis.

[@sarithasaritha.t.r147]

There is always, always a pattern to everything

[@brandonfox9618]

@@MarkusShepherdThey’re not just zeros. They’re “non-trivial” zeros!

[@Jeff-zc6rr]

too bad this isn't the riemann zeta function. This is the analytical continuation of the zeta function, or what we call the functional equation., If you try to put the zeros of the riemann zeta function into the actual riemann zeta function it does not go to zero.

[@SidneySilvaCarnavaleney]

¿Qué impacto causaría si afirmo que he encontrado el número primo más grande y más pequeño encontrado en todo momento, ya que la "Hipótesis de Rielman ha perdido toda su fuerza, ya que afirmo que algunos números no son primos"? Estimado noble amigo de este sencillo canal, con mi respeto a los profesores, alumnos y amigos de este sencillo canal, les reportaré algo muy intrigante sobre estos números primos, con un simple PA (Progresión Aritmética), puedo decir con total veracidad, demostrando científica y matemáticamente que los números que citaré a continuación no son primos, y los primos gemelos no existen: 2; 19; 41; 59; 61; 79; 101; 139; 179; 181; 199; 239; 241; 281; 359; 401; 419; 421; 439; 461; 479; 499; 521; 541; 599; 601; 619; 641; 659; 661; 701; 719; 739; 761; 821; 839; 859; 881; 919; 941; 1019; 1021; 1039; 1061; 1181; 1201; 1259; 1279; 1301; 1319; 1321; 1361; 1381; 1399; 1439; 1459; 1481; 1499; 1559; 1579; 1601; 1619; 1621; 1699; 1721; 1741; 1759; 1801; 1861; 1879; 1901; 1979; ¿Y cómo sería la hipótesis de Rieman, si estos no son primos? Al tratarse de un descubrimiento innovador en el Universo de las Matemáticas, los enunciados de épocas pasadas quedan nulas, dice el autor de la obra "Un atrevimiento del pi ser racional", Sr. Sidney Silva. Dentro de mi obra "La audacia de π para ser racional", demostrando Matemática y Científicamente que es un número Racional e Irreversible con una fracción de números enteros.

[@ゾカリクゾ]

??? This looks AI generated

[@MatveiPB8]

Cursos de la Corporación Andina del Fomento en Economía, Edtoy subiendo el curso 0 para el programa del CAF

[@drakanDS]

music?

[@mihaleben6051]

>looks at graph >crosses zero multiple times So uh...

[@momomomomomo842]

I can't comprehend clearly 'cause the first variable 0 is not on the critical line.

[@HL-iw1du]

I think this video has the real part of the input of the zeta function fixed at 1/2 and the imaginary part of the input increasing as time goes on. The video displays both the real part and the imaginary part of the output of the function corresponding to the input at any given time.

[@ffhashimi]

This is really amazing; and it's very useful ..I can't understand the process from the script so I have some questions : I understood from your post that you add time as variable to the 1/2+yi 1- why did you start with value 1.460.. 2- did that mean you used the first 200 zita zeros? will you explain more how did you create this great animation? and thanks for this animation

[@MarkusShepherd]

The script essentially calculates the values of zeta for 1/2+0i, 1/2+0.1i, 1/2+0.2i, ..., 1/2+14i, ..., 1/2+200i. At each time step this new value will be plotted while the old values "fade". Hope that helps!

[@Meuszik]

how did you create an animation like this?

[@MarkusShepherd]

Sage. I wrote a few comments on the video here: www.riemannhypothesis.info/2016/04/visualising-the-riemann-hypothesis/ it also has a link to the script :-)

[@jennyone8829]

🎈

[@paulthompson9668]

Markus, watching the spiral makes me sick to my stomach. There's just something gross about it, like playing poker with people who don't know how to fold, and ending up losing every once in a while to hands that no one in their right minds would play after a high pre-flop bet.

[@MarkusShepherd]

Paul Thompson you're welcome!

[@paulthompson9668]

Haha Markus. Now I wonder if it would be possible for you to do an animation of the Riemann Zeta function as the values go up all the lines (simultaneously) between 0 and 1. Would we see an interesting paintbrush- like spiral, or would it be complete chaos?

[@MarkusShepherd]

It would certainly be possible and probably not that difficult - I might try it one of these days, or you could try too if you want to get your hands dirty on the Sage code I posted here: www.riemannhypothesis.info/2016/04/visualising-the-riemann-hypothesis/ Using "all the lines" would probably too much to see anything, but with the right selection I'm sure it'd be very interesting!

[@paulthompson9668]

I haven't programmed in Sage before. Is it possible to add comments to the code? If so, could you put in comments in the appropriate places to let me know where I would have to add code to make it do the animation I asked about? I imagine we would need a nested for-loop.

[@stephankuerner315]

I cracked it you lie completely wrong

[@MarkusShepherd]

Sorry, what's there to crack and who is wrong where?

[@MarkusShepherd]

Sure...

[@stephankuerner315]

Sure ... yes - the Dax or the Lotto can be easily calculated and this shows that it is fraud. Because it is to be proved by this - it can not be found (I believe)

[@willyou2199]

i dont think he knows what he's saying