Visualizing the Riemann zeta function and analytic continuation

zeta function. This is one of those objects in modern mathematics that Many of you have heard, But who can be really hard to understand. Don't worry, I'll explain that animation that I only saw in a few minutes. A lot of people know about this job Because there isn't a million dollars away Exit prize for those who can know When equal to 0.

There is an open problem known as Riemann hypothesis. Some of you may I have heard of him in the context Divergent sum 1 + 2 + 3 + 4 … on and on up to endless. You see there is a feeling in it equals -1/12, which looks like Meaningless if not an obvious mistake. But A common way to determine what this equation is He is actually saying he uses zeta Function. But like any math model Enthusiast who started reading this The reviewer knows his definition of this one An idea called analytic continuation who It has nothing to do with functions with aggregate values This idea can be frustrating Opaque and unintuitive, so what I like To do here is just to show you all that This zeta function actually looks like Explain what this idea is from Analytical continuation in visual and way more easy. I suppose you Knowing complex numbers and which You are comfortable working with them, and I am inclined to say that we should know Calculus since the analytic continuation is It's all about derivatives but on the way I Planning to deliver the things that you think you are It might actually be fine without it.

And therefore jump right into it Let's define what this is zeta is a function for a given input where we I usually use the variable s 'function is 1 more than for s" (which is always 1) + 1 over 2 to s' + 1 over 3 to In the picture '+1 more than 4 to the picture' on and on and on Summarize on all natural numbers. So for example let's say you plug in A value like: s = 2 you get 1 + (1 more than 4) + (1 more than 9) + 1/16 and as You keep adding more and more The inverted squares is only up to Approaching pi squared occurs more than 6 . Which about 1,645 there is too Pretty reason why pi appears here And I might do a video on a later date But this is just the tip of the iceberg Why is this a nice post. You can do the same for the other s' input like three or four and Sometimes you get other interesting Values ​​and yet feel everything Very reasonable you collect Smaller and smaller quantities and these Amounts approaching some number…great, no The madness is here! But if you are to read About it you may see some people say That zeta of 1 equals negative -1/12 But given that amount is infinite This does not make any sense…

When you Raise each semester to negative 1 Flipping each part you get 1 + 2 + 3 + 4 on over all natural numbers and Obviously not close to anythingقترب Definitely not -1/12, right? Like any mercenary looking at Riemann Premise known this function is said يقال To have trivial zeros in even negative numbers So for example to mean that zeta Negative 2 = 0, but when you connect At -2 they give you 1 + 4 + 9 + 16 days and a day, which again It obviously doesn't come close to anything Less 0, right? Well we'll get to negative Values ​​in a few minutes but to the right Now let's just say the only thing that It seems reasonable This function only meaning when the s 'is greater than one which is when this amount converge yet she simply does not know for other values.

Now though he said it was Bernhard Riemann To some extent from AB to complex analysis Who is studying which jobs We have the complex numbers as input and outputs. So instead of just thinking about how This amount takes the picture number "on the riyal Line a number to another number on real number line His main focus was on understanding what Happens when you plug in a complex value for s, so for example maybe instead of Connect 2, do plug in 2+i Now if you've never seen an idea Raise the number to the power of Complex value you can feel kind of Strange at first because it's no longer It has to do with recurring Multiplication but mathematicians found That there is very nice and very The natural way to expand the definition The foundations behind being familiar with it Territories of real numbers and in The realm of complex values.

It's not super Crucial to understanding the complex foundations I'm going to share it with this video but I think it would still be nice if only we Summarize the gist of it here The basic idea is that when you write Something like half to strength Complex number you divide it as Half to afternoon part real half To the pure imaginary part we are in good shape on Half on the real part there is no Issues there but what about raising Something for a pure imaginary number? Well the result will be some Complex number on the unit circle in Complex plane as I would like to pure imaginary input walking up and down dummy line output output He walks around this circle alone For a base like half the output It walks around the circle of unity somewhat Slowly but to base this further apart like 1/9 then as you would like This input walk up and down Fake corresponding output axis Will wander in a circle of unity more quickly.

If you haven't seen this And I was wondering what on earth is this Happening I left some links to good Resources in description for here له I'm just going to go ahead What without a reason. Home Takeaway Is that when you raise something like this 1/2 to the power of 2 + i which is half squared times half of i that half on the first part is It will be on the meaning of the unit circle It has one absolute value. so when You hit her it doesn't change The size of the number it only requires that Fourth and spin somewhere. Even if you connect 2 + i to zeta function one way of thinking What you do is to start with all of Raised terms to a power of 2 which You can think of is piecing together Lines of length from inverted from Squares of numbers that as I said Before CONVERGES to pi² more than six Then when you change this input from the two By 2 + i each of these lines gets Manage some amount but the most important The lengths of these lines will not change So the amount still converges just do Even in a vortex to some specific point on complex plane.

Here, let me show what It appears when the input is different Liked with this yellow dot on A complex plane where this sum is a vortex Always shown The value of zeta converges from the picture What this means is that zeta (s) knowledge Since this amount is completely infinite reasonable complex function as long The real part of the input is larger From one input s meaning 'Sitting somewhere In this right half of the plane is complex Again this is because it is the real part From the image that determines the size of each A number while only the imaginary part Dictate some rotation.

So far what I want To do is visualize this job that Takes in the input on the right half of the Complex plane and spits outputs Somewhere else in a complex plane Super cute way to understand complex functions to visualize them as Transitions that means you look at All possible input function and Just let it move over to The corresponding output … for example let's Take a moment and try to visualize Something a little easier than Zeta function: say f(s) = s² When you plug in s = 2 You get 4 so we'll end up with the gif that dot over two more than dot over four When you plug in -1 you get 1 even The point here in negative 1 is It will not end moving more than one point In 1.

When connecting i by definition square of it is -1 So will have to move over here to Negative 1 Now I'm going to add on More colorful grid that is just Because things are about to start moving And it's kind of nice to have something To distinguish between grid lines during those movement. From here I say to Computer to move every single point on This network for more than her interview Output under and function (s) = s² Here's what it looks like I can be a lot to take in so I'll go Go ahead and run it again and this time Focus on one of the marked points and Notice how he was moving more than one point corresponding to her yard.

It can be A little complicated to see all of Point moves every time but the bonus Is that this gives us very rich Picture what a complex function Actually do this all happen in Only two dimensions…so back to Zeta Function we have this infinite sum that is a function of some complex number of the image We are well and happy about Connect the values ​​from the image that are real part is greater than one and get some Visible output via convergence Spiral cell so to visualize this Job I'm going to take part Grid sits on the right side of the The plane is complicated here where the real Part of numbers greater than one and I'm gonna tell the computer to move all Point of this network for the occasion Produce it really helps if you add a few More grid lines around one number Since that area gets stretched out calm down a little Well first up of all let's just Appreciate how beautiful it is I mean damn it doesn't make you want To learn more about complex jobs You have no heart.

but also this Network diverted is just begging to be Expand a little for example let's Highlight these lines here which Represent each of the complex numbers With the imaginary part i or -i after Transform these lines make like Beautiful arches just before you suddenly I stopped Do not you want just because you know continue Those braces actually you can imagine how Some modified version of the function With a definition that extends to This left half of the plane may be Able to complete this picture with This is so beautiful Well this is exactly what mathematics Work with complex jobs do! they continue to function outside The original domain where it is now known as Soon we branch on the input to where The real part is less than 1 this An infinite amount that we originally used to Defining a function does not make sense After that you will get nonsense like add 1 + 2 + 3 + 4 over infinity But just looking at this turned Copy of the right half of the plane Where some does not make sense it's just begging us to expand the set of points that was considered as input even if This means defining durations Post somehow that not Necessarily use this amount of course that Leaves us with the question how it will be You know that post on the rest The Plane? You might think you can extend it Any number of ways you would probably set out for An extension that makes it a point to say…

s = -1 moves over -1/12 but you're probably bobbing on Some extension that makes it land on any Other value I mean once you open yourself On the idea of ​​specifying a job Differently from the values ​​outside this Unsupported field of convergence On this infinite sum your world Oysters can be any number of Extensions right? Well not exactly I mean Yes you can give the child any sign and Have them extend these lines ie which way but if you add on restriction That this new extended functionality has to be a derivative everywhere you settle for us In one and only one extension possibility I know I know… You said you wouldn't You need to know about derivatives for this Video Even if you don't know calculus You probably have yet to learn how Interpretation of derivatives of the complex Jobs but fortunately for us there So beautiful geometric intuition that We can keep in mind that when I say the words Like it has a derivative everywhere here to Show you what I mean Let's look back at That f(s) = s² example Again we think of this task as a Shift moves each point of the image Plane complex for more than s² points For those of you who don't know Calculus and you know that you can take Derivative of this function in any given Input but there is interesting property that transformation which Turns out they are related and approx Equivalent to that fact if you look Which two lines in the input space that Intersect at some angle and look What to turn into after Shift they still intersect each other at the same angle.

Lines may get curved and that's okay but The important part is that the angle that intersect has not changed This applies to any pair of lines to choose! So when I say a job has Derivative everywhere I want you to Think about this angle preservation Property that at any time two lines Intersect the angle between them still Unchanged after switching on At a glance this is easier than estimating before Note how each of the curves that for Grid lines turn into still intersect each Some are at right angles. complex functions which has a derivative everywhere Called analytics so you can think of this Term analysis as angle meaning Keep the Muslim I'm lying to A little bit here but just a little bit to Slight caveat for those of you who want The full details are that the input where Derived from the 0 وظيفة function Instead of an angle it is preserved Get hit by some true, but these points are By far the minorities are almost all Input to Angles of analytic functions Is retained So when I say analytical you think angle Maintain I think that's fine intuition to be Now if you think about it for a moment This is the point that I really want You to appreciate this is very Restricted property angle between Any pair of intersecting lines that remain the same and after pretty much any She works there which has a name Show that field analytics Composite analysis that helped Riemann to Created in its almost modern form Totally about making use of the characteristics Analytical functions to understand Results in patterns and other areas Mathematics and science.

Zeta function Determined by this infinite amount on The right half of the plane is analytic Notice function how each of these curves To turn into grid lines still They intersect each other at right angles So surprising fact about the complex Functions is that if you want to extend Analytical function is out of scope Where it was originally defined for example This function extends zeta in The left half of the plane then if you Long new job required Analytical still that is it still Keeps corners everywhere It forces you into only one possible Extend if it exists at all It's kind of like an infinite continuous The puzzle for this requirement of Keep angles guide you to one and Only one option how to expand it This process relates to analytic expansion function in the only possible way that Still called Analytical as you may have it I guessed "Analytical Persistence" for this How full function of Riemann Zeta is Knowledge of image values ​​on the right Half of the plane where the real part is Bigger than just plug them in This sum and see where it intersects and That convergence may seem some Kind of vortex since lift all of These conditions to complex power have Rotating effect every one then to The rest of the plane we know that there Having one and only one way to extend This definition so that the function Analytical will remain that so is that Still keeps corners in every One point so we just say that through Definition of the zeta function on the left Half the plane is whatever Extension happens to be that is Correct definition because there is not only One possible analytical continuation I noticed this is a very implicit definition It says just use this solution The puzzle that through more Abstract derivation we know must exist But it does not specify exactly how solve it Mathematicians have a very good understanding Apparently this extension like but Still some important parts of that mystery A million dollar mystery actually let's Actually take a moment and talk about Riemann hypothesis The problem is a million dollars Places that equal this job Zero turns out to be very important that he is Which points get assigned to the original After switching one thing we I know about this extension is that Even negative numbers get map to 0 this Trivial zeros are usually called The name here stems from a long time The tradition of mathematicians to connect trivial things when understood Quite well even when the fact is Not at all clear from the start we We also know that the rest of the points That gets you on the map to 0 sitting somewhere in this The vertical strip is called the barbed tape And the specific status of those Non-trivial ciphers encode surprise Information about prime numbers it Actually great interest why is this Post carries a lot of information about Primes and I definitely think I'll make A video about it later but right Now things are long enough that I'll leave It does not explain the Riemann hypothesis that All these non-trivial zeros sit right In the middle of the tape on the line s numbers whose real part is Half of this is called critical The line if true it gives us Understand noticeable narrowness of the pattern Prime numbers as well as many more Patterns in mathematics stem from this so far Even when zeta shows what function Looks like I might just show what you're doing To a part of the network on the screen This kind of under sell her So complex that if I highlightسلط This cash line and application Transformation may not seem to Cross the original at all but used with The version turns more and more This line looks like Notice how it goes through Number zero many times if you can استطع Prove that all zeros are non-trivial Sitting somewhere on this mud math line Institute gives you 1,000,000 dollars I was also to prove hundreds that Did not have thousands of modern mathematics results Which already proven depends On this hypothesis being true Another thing we know about this Extended function that map in Point -1 over negative -1/12 And if you plug this in originally The bottom line seems to be saying 1 + 2 + 3 + 4 On and on up to equal to infinity -1/12 Now it may sound disingenuous to still call this is the amount Since the definition of zeta Function in the left half of the plane This amount is not directly defined Instead it comes from an analytical point of view Continuation of these spontaneously out of range where the solution intersects The puzzle that started on the right Half the plane that Dick said to I admit that the uniqueness of this Analytical continuation of the fact that The puzzle has only one solution Very suggestive of some quintessential The relationship between these values ​​extended The last original amount of the animation This is actually very cool I Guys will show you what Derived from the zeta function theories Like but before that does not matter to me because Let you guys know who made this video possible First of all there are viewers Like you support directly on patreon This video was also private Partially supported by which Availability of audio books and other audio Use of virtually audible reality materials Every day I have some time now if You don't have to already listen to Literature as one of your habits either That's why you're hopping or jogging Or cooking or whatever can be real Life changer one book especially good واحد That I'm listening recently are the art Learning by Josh Weitzkin that got Because of a very sure recommendation My brother and it's one I think you Guys would love Josh Waitzkin a lot The author was a national chess champion Throughout his childhood and later Life took the martial arts Tai Chi Chuan and Become world champion in a few Years That guy knows what it takes to learn And I found a lot to say about Chest and Taichi translate too Useful for learning mathematics as well Especially what he says about starting With the end of the game you can listen to The Art of Learning 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