#### Am Vor 2 years

This video provides a basic introduction to the Riemann Hypothesis based on the the superb book 'Prime Obsession' by John Derbyshire. Along the way I look at convergent and divergent series, Euler's famous solution to the Basel problem, and the Riemann-Zeta function. Analytic continuation of the Zeta function is discussed as well as the sum of the positive integers. Finally, the Riemann Hypothesis is stated and briefly discussed.

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## KOMMENTARE

## Robert S

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Vor 20 Stunden## Василий Суриков

wait a minute, but where is the solution? should I expect a second part? didn't find in your playlists 😭

Vor Tag## Ynse Schaap

I hate jenga 😒

Vor 2 Tage## ll jiminn

Thank you!

Vor 3 Tage## GEO

Is there a term of S(x) missing at the top at 22:32 or am I dumb? Lol

Vor 6 Tage## Stoyo Hubchev

what about zeta(0)? from the equation from the analytical continuation zeta(0)=zeta(1-s) (lefthand side) => s=1. substitute on the righthand side and it becomes zeta(0)=0.zeta(1) but isn't zeta(1) infinity?

Vor 7 Tage## andrea sartori

https://www.youtube.com/watch?v=PfSfjUuKnoU

Vor 7 Tage## Lizard King

Watch at 1.5x speed to avoid the infuriating pauses/slowness of delivery!

Vor 8 Tage## Brian Joseph

Can anyone explain the derivation of the second green formula he shows @ 13:02? He says Euler just imagines this, lol. I don't see how it equates. Thanks!

Vor 8 Tage## Mr Goldie

The Riemann hypothesis is a fractal.

Vor 8 Tage## MAD VOICE

Simple and sweet bro you are super

Vor 13 Tage## Mars Truth

Why the harmonic series and not a geometric series, say r = 1/2? Is there a theorem to this effect?

Vor 13 Tage## Peter Gacs

Nice exposition, but in my view it misses the more important part: explaining the connection to the distribution of primes, which gives the hypothesis its main interest.

Vor 18 Tage## Seth Connor

The best explanation I have ever had the pleasure of seeing thank you for sharing

Vor 18 Tage## Lawrence Shuda

W.O.W.

Vor 18 Tage## Hjbjuviyg Hjvhjvjy

But if you assign a mass to each of the blocks, it will topple at 2.5.

Vor 19 Tage## Kihm Jones

The answer is so obvious .. when the center of mass is beyond 1/2 length of the initial block the tipping point occurs .. WTF are you doing

Vor 20 Tage## Narf Whals

That is why we decrease the overhang with each block. The center of mass never goes beyond the critical point. That is also why you need a rapidly increasing number of blocks for a given length.

Vor 19 Tage## Daniel Kanewske

I appreciate your approach. Unless I'm mistaken your definition of sin(x)/x as a factored polynomial doesn't meet the f(0) = 1 criteria because f(0) is undetermined. You will need a more robust definition of your function, i.e., f(x) = sin(x) for x > 0 or x Your explanation concerning S(x) = 1/(1-x) is misleading. The infinite series is only equal to the expression 1/(1-x) when the infinite series is defined. For all values such that the sum is infinite the algebraic manipulations are invalid, example infinity = 1 + x*infinity, true or false? The question itself is ill-posed.

Vor 21 Tag## Ariel Galindo

https://youtu.be/MCNR8Euw9rc

Vor 26 Tage## Dog Meat

These are lectures? i've been watching them for fun.

Vor Monat## David Sansom

Absolutely the best and clearest explanation and understanding of Riemann zeta function and Riemann Hypothesis I have ever seen. Can you make a video on complex analysis and geometric algebra. Also how about Einsteins Field Equations and Gravity. Super and exciting well done!!!

Vor Monat## Physics Explained

Thanks!

Vor Monat## Flash213

So it’s the reverse of making a retaining wall mmmmmm

Vor Monat## Jan Fanta

Erweiterte Vermutung Riemanns , Fantas Vermutung. Die Anzahl der Primzahlen ist bekanntlich unendlich , aber siehe da , sie sind gleichzeitig eine Untermenge der natürlichen Zahlen , die wieder eine Untermenge der ganzen Zahlen bilden , wobei die ganze Zahlen eine Untermenge von Rationalen Zahlen bildet , und diese eine Untermenge von reellen Zahlen darstellt. Die Mathematiker haben aber die Kühnheit zu behaupten , dass die Anzahl der Primzahlen unendlich ist , was bedeuten würde , dass die unendliche Anzahl der Primzahlen , also das Unendliche eine Untermenge einer wesentlich größeren Anzahl an Zahlen untergeordnet ist, respektive es gäbe mehrere unendliche Zahlen-Kategorien , mit anderen Worten , es gäbe Unendlichkeit , derer Untermenge unendlich ist , was , wie die ganze Mathematik ein Absurdum ist. Was man aber physikalisch mehrdimensional einwandfrei beweisen kann , ist die Tatsache , dass alle mathematische Zahlen , die wir nicht gefunden , sondern erfunden haben , als eine 0 - Stelle in kritischen Streifen liegen. Gruß 12 D Physik ohne Ränder Davids.

Vor Monat## hope dies

I am far too stupid for this.

Vor Monat## not me

😐

Vor Monat## Pete Rembranch

Only suitable for people with a strong maths background. Laymen will have to look elsewhere.

Vor Monat## ja maguire

1:29 Why do you call this the "Harmonic Series", when it's not a series, but really just a single real number?

Vor Monat## Sam Anderson

It is the well founded theory that the Romulans are NEVER going to let us out of these damn mines!

Vor Monat## Sean g 137

here is the video where I demonstrate the use of a formula to predict primes with lottery winning odds that I just cracked while working on the Riemann Hypothesis. https://youtu.be/wASftNDzBuo The video is an hour long with the explanation and background but the actual demo of the calculations are at minutes 10, 22 and 43

Vor Monat## Ultrametric

Why is this in a physics website? BTW there is nothing interesting whatsoever mentioned about the main topic. Just a video based on Wikipedia. So why are you doing this? So people will think you are on top of things?

Vor Monat## steve Zara

I know it’s been two years, but thanks for such a beautiful and clear explanation.

Vor Monat## Michael Ashcraft

Who is a Rieman?

Vor Monat## Bob Laughlin

The narration is impeccable, especially since it was done back in 1961 by Mick Jagger when he was studying at the London School of Economics.

Vor Monat## Paramount Technical Consulting LLC

Love it! "Part of a function defines part of a domain.." Parity and Symmetry! Math is the most fundamental way we interact witht he universe. There is no solvable function that is not applicable to a physical phenomenom, no matter how abstract, and anything that is not solvable cannot exist. And color is defineded by wavelenths and ptich is described by harmonics; the painter, the poet, the musician and the mathematician are all touching the "soul" of the universe from different viewpoints. Life is good!

Vor Monat## Zero Ryoko

Why do you do that @2:45? Why re-write it? You wanted to compare the divergent series 1/2 + 1/2 ... with the harmonic series 1/2 + 1/3 ... Yet for no reason I can fathom you change the divergent series to fudge it, making it seem that the harmonic series diverges when it clearly does not! the first term is equal, the second term; 1/2 vs 1/3 clearly shows that he harmonic term is less and as it continues the harmonic series gets less and less. therefore clearly the Harmonic series *does* *not* *diverge!*

Vor Monat## Sean Kirby

I love math, but I'm no mathematician. However, I believe a lot of math is flawed, for the simple fact that they continue to rely on 'infinite' types of problems. I do not believe there is an infinite amount of numbers between 1 and 2, etc... There is a limit, no matter how small you go, you will reach a point where you can't go any smaller. This renders the infinite theories as flawed. I know you can't prove that infinity in this way exists, as I can't prove it doesn't. It's just a nice theory! I don't believe the universe is infinite either - I believe it has a limit, as are the number of stars, etc... Infinity is a nice term used when nobody knows how many there are.

Vor Monat## Rafi Rodríguez

It seems that the expression you have used to factor x from S(x) at 22:27 should be S(x) = 1 + x(X + x^2 + x^3 + ...) not S(x) = 1 + x(1 + x + x^2 + ...). Your factoring actually redefines S(x) as 1 + x + x^2 + x^3 + ... which is not your originally presented function S(x).

Vor Monat## The Swartz

You've never played Jenga with my wife... Which brings up the question... when are you doing a video on Chaos Theory??

Vor Monat## jim twisted

At 13:15 how did you go from the 2nd to the 3rd line?

Vor Monat## Michael Price

"And he didn't stop there, in fact he went as far as calculating 1/n^26" Because why not?

Vor Monat## Meena Deshpande

Hi This is Abhijeet Deshpande and.... This is how to understand the theorem.... Points: 1.) From 1 to 100, calculate the number of odds and evens 2.) Now for every single of the odd and even numbers, measure and write down the number of steps, for both to go to the number 4. 3.) For both the odd and even numbers, calculate individually as below, a.) Add the number of steps to get a total of both odd and even b.) Get a total of odd / even numbers by addition i.e. a.) How many numbers are odd and even b.) And what the the sum total of odd and even by addition c.) What is the sum total of odd and even by division d.) What is the sum total of odd and even numbers by substraction 4.) Divide the number of steps with the number of odd / even numbers wiithin 1 to 100 5.) Now upon fiding the value of the division of both odd and even numbers, Use the above results of calculations to calculate with the results to determine the base structure or the point of average c divisions or calculations, where both the calculations of odd and even align, And Voila, you have a symmetry of calculative set of equations that would determine the results of any similar supposedly unsolvable equations. These equative calculations of mine can also solve the problems of Rieman hypothesis of prime numbers as well. As such I am eligible to win the seed of Clay Institute for of and towards the same. 3x+1, Rieman Hypothesis © Abhijeet Deshpande, 2021

Vor Monat## David Littleboy

Finally! Now I know what Lou Reed was singing about when he sang "I am tired, I am weary, I could sleep for a thousand years...". He clearly was trying, and failing, to solve the Rieman Hypothesis.

Vor Monat## IrelandVonVicious

From the looks of it we use the wrong math to describe this. It is outside our current structure. Similar to the Romans not having a zero and the debate with the Greeks of whether or not zero exist and should be included at all. I believe that the answer to cosmology lies within this problem/solution.

Vor Monat## michael york

So when God, a self contained sentient solid sphere, needed to omnidirectionally reconstruct His Geometry to a hollow sphere, the convergence point would have to have been less than 1 to achieve final balanced constituency without final disassociation. Over some finite expansion, with a finite timescale to achieve balanced equilibrium. Considering the origin of the "blast point", and the resultant redirection of energy necessary to instantiate this redirection of information, a Sentient Consciousness would have been faced with calculating the gradation of frictional loss needed to achieve bounded infinity. There is a convergence, but no level of calculation could ever uncover it, for it is too vast a number for all human life to have ever, even with quantum computers, and some absurd number of lifetimes, to ever define it as fixed known origin constant. The energy transference necessary to go from a solid sphere, to a hollow sphere had to have a ratio which ensures a rebalance point, a harmonic, was achievable. The God Calculation. How Fine a gradation of calculation is necessary to search "within" an infinite set of information? God found a way to sift out the one smallest, within the infinitely largest. Where is the exact center of and ever expanding sphere?

Vor Monat## michael york

This is equivalent to Pi, which displays the error involved, in constructing a curve, using infinitesimally smaller line segments, which can never be achieved.

Vor Monat## Ian Consterdine

Brilliant exposition, thank you. Have you also released a talk on how, exactly, the Riemann hypothesis relates to prime numbers?

Vor Monat## RogerWKnight

Here is a question that has never been answered: Of what practical value is all of this business of trying to predict the distribution of prime numbers? Pre-Metric systems of units tend to use the opposite of primes: 360, 1760, 5280 and so on. Numbers that have a large number of integral divisors, or factors. We see 360 as the degrees in a circle, shakus in the pre-Metric Japanese cho, and the home run in feet in baseball as well as the length of the football field in feet. 360 and multiples of 360 are popular in computer screen resolutions. If we divide the Moon's diameter into 1/3, and call that the Lunar Third, we find that the Earth is 11 Lunar Thirds in diameter and that the Sun is 1200 Lunar Thirds. In English miles, the Lunar Third is 720, which is 6! The diameter of the Earth is 7920 English miles, which is the number of inches in a furlong. The diameter of the Earth in furlongs is 63,360, the number of inches in a mile. 720 is the number of minutes in the 1/2 day represented by the circular clock, 1440 is the number in a full day. The number of seconds in a day, 86,400, is 1/10 of the number of English miles in the diameter of the Sun, 864,000. It is almost as if the Medieval English had as good an idea of the sizes of the Earth, Moon, and Sun as the French Revolution had of the size of the Earth. At least 10 and powers of 10 are not prime numbers! Pi being estimated as 22/7 (good for practical calculations such as made by an engineer) might explain the furlong being 220 yards and the Egyptian Royal Cubit of Thoth being 7 palms. But I say again, beyond 11 and 13, what is the practical value of all of this research into primes?

Vor Monat## DeathBringer9000

pi squared over 6 is a hexagon

Vor Monat## Ron Walker

MATHAMATICAL EXERCISES WHICH HAVE NO AESTHETIC APLICATION, AS IN A GREEK TEMPLE BORE ME TO DEATH. IT IS LIKE PUTING THE MUSIC OF BACH INTO A MATHICAL FORMULAT. IT WAS JUST HIS SENSATBILITY.. DITTO MOZART ETC.

Vor Monat## Diktakt

22 39 !

Vor Monat## Mintu Saren

Are you belief in Hindu @ Vedas.

Vor Monat## Felixinimon I

Do you play counter strike :????

Vor Monat## What about numbers and notes ?

The lower boundary of an expression deduced from Dirichlet Eta function is equivalent to the impossibility of symmetrical Riemann zeros on either side of the critical line. https://www.youtube.com/watch?v=EMqnSEhvMs4&t=6s

Vor 2 Monate## Saving humanity from extinction, anyone?

Never even heard that word, "Jenga" before, let alone Jenga blocks. Where in the heck are you from, guy? Lll!

Vor 2 Monate## JKItsMeAlGore

At 4:30. Hard to believe that the measurements for that number would be larger than the observable universe…. That surely cannot be right can it?

Vor 2 Monate## Hermanto Surjani

According to my calculations, it would be bigger than the observable universe by a factor of five thousand.

Vor 2 Monate## Tadesan

The first question I would ask is; where’s the bathroom at?

Vor 2 Monate## Dr Ajay

Can you evaluate this solution https://youtu.be/8zelcgt7xVQ

Vor 2 Monate## Scott Reynolds

So the solution to the hypothesis is glue.

Vor 2 Monate## David Wilkie

The even form is in the picture plane, and convergences in the vanishing-into-no-thing of 2-ness, ..and the odd linear version in z-axis=normal orthogonality in i-reflection containment all-ways aligned to root zero-infinity, sync-duration 1-0 probable connectivity, ie zero difference @Singularity.., "infinite trivial-> zero-infinity" => flat-space ground-state No-thing-defined eternally. Perhaps Analytic Continuation is not applicable to this Polar-Cartesian coordination representation because of the transverse trancendental e-Pi-i sync-duration connectivity function, under which condition of omnidirectional-dimensional logarithmic interference positioning, unity is orthogonal to the 2-ness picture plane as observed in WYSIWYG QM-TIME Completeness Perspective Principle. (Observable state-ment, proof-disproof, self-defining condition) The 1-0 probability dominant numberness dominance sequences of existence in the zero-infinity time-timing Quantum Operator Logic Field Modulation Mechanism Singularity inside-outside holographic positioning is the Observable Natural Philosophy holistic POV, Temporal Actuality Conception.

Vor 2 Monate## 06howea1

Haven't they built an ai proof resolver

Vor 2 Monate## lobsterfork

This is the explanation I have seen so far. So much so, you have inspired me.

Vor 2 Monate## The Old Geeks

I feel dumber and smarter then before at the same time

Vor 2 Monate## Bashir Hemadi

that was the best explanation

Vor 2 Monate## bobby dorou

Have never ever heard of this before. I am therefore so ignorant.

Vor 2 Monate## Bonnie Massaro

This is so silly, the problem is the solution itself. It is a hypothesis until proven untrue. And it is based on a straight line. Therefore, untrue. No longer science. This is where Good ol' Riemann went wrong, there are no straight lines in physics, just a wobble with energy / eather in the middle, that is your 1/2 and the reason no-one can figure it out, the hypothesis includes an assumption. A straight line. And there is absolutely a pattern in the prime numbers, follow the space between 7s. Bet bitcoin gets nervous. Good trick all, way to keep a bunch of brainiacs busy trying to figure something out, that within itself is not based on truth. Beware of assumptions!!! Indeed.

Vor 2 Monate## Rubens Cabral

the primes are "vortex" numbers without the number "0" it is in the center in the "vortex" the numbers go back and forth with their opasts 7 and opposite of 3 or 6 opposite of 4 also 8 opposite of thanks.

Vor 2 Monate## Rubens Cabral

Olá sou do Brasil a fórmula dos não primos automaticamente você descobre os primos são 3+3+3+... ao infinito e os 7+7+7... ao infinito e os quadrados perfeitos impar menos com final 5 exemplo 9×9.11×11.13×13..the three infinity of the same size thank you.

Vor 2 Monate## Colin Dant

There's a term missing in the S(x) series Domain Stretching section.

Vor 2 Monate## geogy kanicheril

Wonderful explanation. A typo at 13:20( the third equation is not true, as terms in powers of x higher than 2 are missing...)

Vor 2 Monate## L'Artisto - Kirthi & Nals

Riemann Hypothesis which is in news now, explained in tamil 👇👇👇 https://youtu.be/Ig8Iu0SXAEo

Vor 3 Monate## Alien

Here's how Fermat would solve it https://youtu.be/IyOvM7VeOGw

Vor 3 Monate## Craig Wall

In these trying times- both societal and personal- your thoughtful, calm, and transcendent lectures are as mentally soothing as anything I have ever come across. They are as good as a solo flight in a sailplane on a day with abundant lift and no air traffic- which would be my normal comfort preference. They instruct, inspire, and yet challenge without generating a feeling of wasted effort, even when not fully understood... because they do not discourage, but rather reward the simple act of paying attention. _Thank you._

Vor 4 Monate## R Hugh Sirius Ph.D.

Zita is Italian macaroni.

Vor 4 Monate## R Hugh Sirius Ph.D.

I only reel out as much of my c*ck to impress the math ladies.

Vor 4 Monate## Manu Blanco Valentín

Great video. There is a small mistake, though in minute 22:30 At the top, when you define S(x) you missed the term "x^1 = x".

Vor 4 Monate## Wolfgang

22:10 There is "+X" missing in the first row.

Vor 4 Monate## brilliantbutlazy 2003

Excellent video, I'm just speed running all your videos right now ;)

Vor 4 Monate## Srinivas Aprameya H.s

Ha singularity in mathematics,good!

Vor 5 Monate## Cenit Magnitud

Is the video's audio choppy for anyone else?

Vor 5 Monate## Muhammad Nada

Great video, thank you.

Vor 5 Monate## Physics Explained

You are welcome! Thanks for the feedback

Vor 5 Monate## Dragrath1

Unfortunately I suspect this hypothesis is subject to Gödel's incompleteness theorems as it seems like using the function itself to try and prove or disprove this theorem is a self referential problem though analogy with the Halting Problem i.e. if you make an algorithm to compute the nontrivial zeros of the Riemann-Zeta function that would halt if a nontrivial zero that doesn't have a real part equal to 1/2 is that not analogous to trying to determine when a Turing machine will halt? I imagine any possible solution would need a greater formalism that extends beyond the complex numbers perhaps an approach based around Quaternions?

Vor 6 Monate## brenda williams

Once you hit the 100 overhang you are at1.9

Vor 6 Monate## brenda williams

It well may be that the overhang is there because you’re not going to get primes dreaming

Vor 6 Monate## Ghristpoher HithchensII

Where art thou?

Vor 6 Monate## real image

私はリーマン予想の 1/2 という数値が発生する事に関して、 The Real number 0 and the imaginary number 0 are spatiotemporally offset by 1. という事が関連しているのではないだろうかと思っています。

Vor 6 Monate## Christian Faust

simply great

Vor 6 Monate## Eanna Butler

11:45 the top expression of f(x) is indeed f(0) = 1, but the lower expression of f(x), f(0) is undefined.. So they're not the same function? And so the lower f(x) can't be used on the following slides has having the required f(0) = 1 ?

Vor 6 Monate## Eanna Butler

Thanks again. https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/1.-differentiation/part-a-definition-and-basic-rules/session-8-limits-of-sine-and-cosine/MIT18_01SCF10_Ses8a.pdf does indeed help explain why f(0)=1, by definition, from reasoning that lim f(x) for x->0 = 1. So I haven't found a hole in the Riemann Hypothesis after all :-D

Vor 5 Monate## Eanna Butler

@Nasser Al Lammah OK, yes I can see lim sin(x)/x = 1, as x tends to zero. But the value of f(0) is undefined. I also know that 0.999 recurring = 1, so from a different perspective, maybe that's why there isn't a hole... I see I need to understand indeterminate forms for 0/0, and l’hopital’s rule.. Thanks again Nasser

Vor 5 Monate## Nasser Al Lammah

@Eanna Butler if we are both talking about the f(x) = sin(x)/x then just plot the function with geogebra or wolfram. It is percectly smooth around x = 0. The term alone as sin(0)/0 is undefined that is true but we talk about function f(x) = sin(x)/x for all real values of x. And for that purpose we have defined the value of that function at zero as f(0) = 1. The term sin(x)/x is just a formula for that function which you can apply for x from (-infininity, 0) and (0, infinity). For x = 0 you use formula where f(x) = 1. And the reasoning behind is what I mentioned in previous comment.

Vor 5 Monate## Eanna Butler

@Nasser Al Lammah Thank you for replying. I get that sin(x)/x approaches 1 for small x, but evaluating sin(0)/0 = 0/0 is undefined? So I don't get how the graph of the function could be smooth... Has to have a hole at x=0? I need to revisit limits I guess! And the thorny subject of division by zero. And other associated ... stuff!

Vor 5 Monate## Nasser Al Lammah

They are really the same, if you analyze function sin(x)/x (plot it with wolfram alpha for example) it is "well behaved", smooth function even for the value x = 0. The way how you determine the function value at zero is that you calculate the limit of it as x approaches zero. And as it is known the lim sin(x)/x for x -> 0 is indeed 1. Try to search on Google for limit of sin(x)/x as x approaches zero. Then the value of this function at zero is "additionally" defined by this limit. Also you have a mathematical proof from the Taylor's theorem - when you calculate the Reminder of the taylor series for sin(x) you find out that the reminder goes to zero for n going to infinity. So it is mathematically proven that the sin(x) and its infinite taylor series are really equivalent.

Vor 5 Monate## Mark Freeman

What makes this problem so compelling, is not just that it is highly coveted proof, but it seems quite straightforward, which lulls mathematicians into a lifetime of work with still no answers. I am sure this problem can be solved, and when it is, we will look at the answer and say, hey that looks fairly straightforward, put that in high school maths books/exams. The children will have no idea that it took hundreds of years to come up with the answer from first principles.

Vor 6 Monate## Mark Freeman

16:30 Homework task

Vor 6 Monate## Wakssbm

Hands down the best video about the Riemann Hypothesis, let alone the slow start. I'm not sure if I was Hilbert, the first question I'd ask would either be that or if P=NP

Vor 6 Monate## Ammar Durghalli

I wish the connection to prime numbers is explained in this video. As it is the full story really feels untold.

Vor 7 Monate## magic

The new finds in the field of prime numbers. The prime numbers form so-called nests of the prime numbers in the fourth dimensions. Please see the homepage www.number-galaxy.eu in the directory "news" and positions: 01.01.2020 3D bordered prime magic squares in world and antiworld configuration 03.02.2021 Projection 3D bordered prime magic squares on critical linie of Riemann zeta-function. This is completely new in the field of the Riemann hypothesis.

Vor 7 Monate## Lightspeed

Zajebiste! (Awesome!)

Vor 7 Monate## Stanley Dodds

It's very important to note that domain stretching as you call it is not unique for real functions, no matter how smooth you require the larger function to be. This has a lot to do with bump functions. The point of complex numbers is that smooth complex functions are very nicely behaved, while smooth real functions are not. You can only extend a smooth complex function in one way. Your example with 1/(1-x) is quite misleading.

Vor 7 Monate## Bob Fisk

Beautifully done! I’ve not seen a better treatment that was aimed at maths mortals.

Vor 7 Monate## Saleri Sinclair

What is a closed form solution?

Vor 7 Monate## RobertMStahl

&c hydrino

Vor 7 Monate## Barnabus Korrum

Was the 1 carried?

Vor 7 Monate## King Him

I solved this just uploaded my notes go check it out.

Vor 7 Monate## Zabir MH Mahdi

make a video on hilbert spaces

Vor 7 Monate