A Direct Proof of the Riemann Hypothesis - Part 1: Background on the Riemann Zeta Func tion

Good evening mathematicians and lay mathematicians and welcome everyone else. My name is Jeff Cook. I'm going to present a series of videos. This is the first part, a little background to a paper that I, Greg, and Dennis have in pre-print right now. It is entitled the direct proof of three-minute hypothesis. Try as best as you can to set aside your suspicions and just enjoy the mathematics that's going to be presented. In the end, the Riemann hypothesis is just a mathematical problem and like many that I'm going to present here that lead up to it, you will see it's pretty easy to get your head around once you understand what's going on. Now we are taking this very slowly. About a year and a half ago, I presented a finding to Greg and Dennis and we've been poking away at it and this paper is the result of that. It is a proof of Riemann hypothesis. It's a direct proof. It's an algebraic proof. It's very falsifiable. We have had a number of mathematicians try to beat it up and to date there are no errors found. We do know that every equation in the proof is correct and now we believe it is time to begin explaining the proof and I don't believe that I would be fair if I didn't include those who were non-mathematicians. Because of that, I am going to explain everything leading up to the Riemann hypothesis including my findings and so that everyone can be on the same page. Experts in number theory as well as anyone with a high school, first year college mathematics background. Okay, I also am going to do my best. Please forgive me if I fail. I'm going to do my best to present this in a completely non-pretentious way. I'm just going to try to make it fun and relaxing and enjoyable. If there's anything fun about mathematics, I'm going to find a way to present it. Not sure if there are. Okay, so a little background of who we are. All right, Dennis P. Allen received his PhD at University of California, Berkeley with distinction. He then went on to as assistant professor teaching calculus, logic, general mathematics, you name it. He's quite an accomplished mathematician and a very smart and nice guy. Greg Volk is equally sharp. He graduated from Stanford. He has always been too humble to tell me but I heard from Dennis that he graduated the top one percent of his class. He has written and edited numerous papers, mathematic papers and physics and has a very good way of tearing apart other people's work and we've had many heated conversations in the past year and a half and I think that's kind of necessary. The heated conversations are all so we can all get on the same page. What are we discussing and how could it be wrong and going over all these. He writes out when he's going over somebody else's work. This is the way he presents it when he's or how he presents himself when editing people's papers is he will rewrite every equation and rewrite it and simplify it and tear it apart and if there's another way to express that equation in a simpler way he's going to find it. Amongst the three of us we have found a way to get this paper or this proof under 10 pages. This was our goal and under 10 steps. I'm going to begin way back at the beginning. When I mean the beginning I mean back all the way to Plato. So that's where the origins of this hypothesis lie. Now where to find the pre-print you can go to Google and do a search ResearchGate Cook Riemann hypothesis and it should be the first link that comes up. You can download the pdf from researchgate.net and if it doesn't come up in Google for you just go to researchgate.net and do a search for me Jeffrey Cook or Dennis Allen. Greg is not on ResearchGate now. Now you do need to be a published researcher to become a member but you do not need to be a member to download this print. We've made it public. All right so it's been up there a few months we're taking this very slowly and feel free to go pause this and go get the paper or after you've done watching this video or the entire series which is going to three or four videos feel free to get the paper. All right let's get started. Oh before we do if you see from my other videos I am a big fan of dark humor. There's no dark humor in my proof or in this presentation but if you are a fan of dark humor and need a little laughter in hard times in the world the world is a very dark place right now. I think the best way to brighten that up is with a little humor. If you're not into dark humor please do not watch this book. It has received some good reviews from top reviewers at Amazon. Grady Harp he's a hall of fame he's a top 50 reviewer. He writes Jeff Cook pounces onto the literary scene with a fascinating first novel that is one of the more interesting concepts in storytelling to grace the bookshelves in some time. I do like fiction. I'm not a big fiction writer. This is the only novel I've ever written but I I'm a big fan of it particularly dark humor. Okay and Daniel Jolly super nice guy he's also a top reviewer. He gave it five stars. He writes Jeff Cook's first novel really stands out from the crowd in terms of both originality and excellent insightful writing. Thank you Daniel. All right I am working on another book. It is a book on physics. It goes also back to the ancients all the way from the ancients to today and some of the highlights at the high points of humankind's research into light. Okay it's a book on physics. I deal with Maxwell, Faraday, you know Euclid. So I got about one chapter left and I hope to have that soon time permitting. All right let's get into it. Part one is just leading up to the Riemann zeta function. So everything leads up to the Riemann zeta function. Well not everything but what is relevant or relevant to the proof. Now this is an unscripted video so bear with the ums and you know all everything else that comes with an unscripted video. I am not very good. I'm much better at reading but I'm going to give it a try. All right so we got some gold miners in this leading up to the Riemann hypothesis. Mathematic gold miners. Okay we're going to begin for the first with the sieve of Eratosthenes. All right so a sieve in gold mining is where you take you're panning for gold and you scoop out the rocks at the bottom of the river and to a certain angle. Gold's really heavy so it's going to get stuck in the sieve and all of the other rocks fall out. So what we're going to do here is we're going to take the natural numbers greater than one and we're going to shake them up a bit and see if we can drop out and leave behind just the gold. All right here we go. Okay so remove every second number after two. Just eliminate them. So we still have two. We have three. But then we got rid of all the odds. We got four, six, eight. All of them just dropped out. We're going to do this now every third number after three. So we leave behind two and three and five. Then it skips up and it gets rid of ten and all of the multiples of actually ten is with the multiples of two. So remove all the multiples of three. All right so now we're going to keep on shaking the pan or the sieve and we're going to remove every fifth number after five and then eliminates all the remaining multiples of five and we're going to continue doing that with every number that is left. Now we're going to go to and we continue the same way to infinity and the gold is what's left behind and these are the prime numbers. All right you know what prime numbers are? I don't think I need to explain them. All right they are the building blocks of all of the other numbers. See so we moved all of the products and we left we're left with the core of the natural numbers. That's the way I like to say it. Okay see that trick? We're going to use it. All right and I'm going to continue this analogy of gold prospecting as we go through here. All right now jump this that's the sieve of I'm not sure how to say that. All right so a bit of a tongue twister for me but um that was done I think about 50 to 100 years after Euclid. All right Euclid was the first one to prove that the number of primes are infinite. Okay so now we're going to jump to the early 1800s. We're just going to skip away so you see I'm not touching everything that was done with it. Let's get way way into the future to the early 1800s and we're just going to begin with some very elementary arithmetic. All right one divided by one everyone knows is one. Oh I'm using team fewer and it's not used to it. One divided by one equals one. Okay something's locked up. You bear with me? There we go. Okay equals one. I knew it did I just you know it's a tricky calculation but one plus one half equals three halves and I'm going to add a third and we're going to add a fourth and we're going to add a fifth to infinity and as you see as we go towards infinity the numerator on the right hand side of the equation is increasing far more rapidly than the denominator and so when we take this out into infinity we're going to end up with an infinite sum infinite value. Not very meaningful other than the fact that this harmonic series is prevalent in nature and so it's not unheard of to have infinite sums. I mean time after all is an infinite sum one second plus two seconds so but manipulating such sums is not very possible and so we want this to converge in some way so how might we get to increase the denominator faster than the numerator so that we can get a finite sum and we could do that by squaring the denominator and let's do that we get this nice value 1.64493 and so on. It's a very beautiful number and this number kind of posed a problem for mathematicians. It's called the basal problem. It was well is there a rational expression? Is there an identity of this number? We look at this in rational form and many mathematicians you know try to tackle this and struggled and failed. All right so along came a very intelligent mathematician and like the sieve we did earlier he had a way he approached mathematics like he was gold prospecting prospecting for gold and and I'm going to keep that analogy going because I think you're going to find it very useful. So let me just explain a little how prospecting for gold works. All right so if you're going out into the high desert and there's mountains off in the distance you're going to eliminate certain mountains that aren't good candidates okay. The red mountains probably not good. They're not going to have the ideal structure or the ideal rocks. They're not you know in this field okay but they're going to be red and they're from a volcano. The gold's going to be buried. You're not going to find anything. So once you find the right mountain then you kind of go along try to start climbing towards it. You find a river and you start panning for gold. Once you find a little gold you move up river and you find a little more gold you move up river and you keep going in this way until you stop finding gold and then you backtrack to where you last find gold and you start climbing the mountain at that spot. So that means the gold is coming down in that area and it's washing out the other. So Euler kind of does that like the sieve of the rot. I'm going to try it. All right hold on. The rot of the knees. Oh my goodness. All right. My name's easy Jeff Cook so it's very easy to say. So maybe that's why I don't practice this. Anyway this is the first gold nugget that Euler found and it's a very beautiful rational expression of pi squared over six. It made him instantly famous because of all the other mathematicians who have tried and failed and he had a proof at first so he thought he solved the basal problem but his proof was not accepted. It took about five years to come up with one that was accepted but the damage was already done. He was instantly famous and so the gold rush begins. Everyone starts trying to throw other powers at this. We would call it a generalization of the harmonic series and let's go throw three into the power of the denominator and we get this beautiful number but we can't find the gold. Okay. The only thing we know about this number is that it is a transcendental number. All right. And that was discovered many years after Euler. Okay. So we look for the power of four in the denominator and very nicely another gold nugget pi to the power of four over 90. What about five? No. No such luck. What about six? Yes. Another one and as we continue through all of the even integers greater than well all the positive even integers we get we know everything about these numbers so that's where all the gold is at this point. Okay. What do we know about the odds? Very little but as you will see and I would say many if not most mathematicians are unaware of this but if you were to tear into Riemann's paper which we're going to get into in a second you will see that all of the value output of this function will involve pi to the power of s over two or except for the evens. All right. So they're going to cancel to just leave pi to the power of s. Don't want to get into that. However since it is not readily known for the odd numbers what rational number if a rational number is a multiple of that we can't express it in rational form. So if you want a really difficult problem to tackle and you think you're up for it go for this one. It's been around since before the Riemann hypothesis event. It's a very difficult problem. Look at those odd numbers put them in rational form and prove it. Okay. So this is called to the generalization of the harmonic series where we are raising the denominator to a power some power and we're going to call that power we're going to give it a name s. Okay. So like I tell my son if you have very little mathematic background you can use this analogy of a function. I tell my son think of a function like a microwave oven. You give it some input that input is time and that microwave oven does one thing typically. It gives some heat over time. That's your output. So you give the function input and you get the function to output something. All right. So our input is s and we give it some value s some value and we should get something out. Now if you if you were to hammer around on this generalization of the harmonic series you will find that well you can't just plug in any old s. You can't plug in one. That doesn't work. That would just reduce to the harmonic series itself. Can't plug in zero for s. You can't plug in any of the negative integers. You can't plug any complex numbers. Okay. You can only plug in numbers where the real part of s is greater than greater than one. Okay. So that is leaves us kind of hanging. A couple of things we might notice about that as we get going. All right. So again comes Euler and he's going to want to do some more prospecting. All right. He didn't tell us really how he well kind of you read his work see a little bit how he got that first nugget. He did the proof. But sometimes when you have when you find something the proof may not represent how you found that. And so that's why I'm presenting this these videos because it's going to lead up to how I found the proof. The proof would not include any of this in these in these first three videos. And so how Euler found the first gold nugget is interesting but we're going to see how he does his prospecting very quickly as we look at this. So he's looking at this and he wants to he's looking for something more. He's looking for gold nuggets just like Eratosthenes. All right. Did I say it right? Help me out if I didn't. Okay. And how's he going to do that? So he's going to take this. He's going to give the left hand side a name. Right hand side. So left hand side. And we're going to call it Zeta of S. All right. We gave the power and the dominator its name S. And we're going to have a function now named Zeta. And we're going to do some prospecting just like like panning for gold like the sieve we did earlier. Multiply both sides by 1 over 2 to the power of S. Okay. And we're going to like the sieve we're going to I'm not even saying his name anymore. You see that? Eratosthenes. I can't do it. Sorry. I refuse. All right. Eliminate the multiples of 2 in the denominator on the right side. Okay. So that's what we just did. We subtract 1 minus 1 over 2 the power of S. And then look on the right hand side. All the multiples of 2 in the denominator are gone. All right. I'm going to repeat this. Eliminate the multiples of 3 in the denominator in the same way we did with 2. And we're going to repeat. Multiples of 5. And as we keep going on into infinity it reduces to 1 on the right hand side. And we're going to solve that for Zeta of S. And we get what is called the Euler product. All right. Euler product is very beautiful because it connects the prime numbers to the natural numbers. You see that? So we have the generalization of the harmonic series is equal to this product on the right over the primes. Like that? It's pretty amazing actually. So now we have this equality. All right. And I'm not saying Euler stopped with it. But he did find the gold. Now along comes another individual. His name is Riemann. Okay. Now he doesn't prospect like Euler at all. All right. If Euler climbs up the mountain and digs for the gold, Riemann turns the mountain upside down and shakes the gold out. Okay. That's just his approach. In fact the magnitude of manipulation that Riemann does it's just it's hard to comprehend how that fits in all one brain. Okay. But he does lay out some very beautiful results with what he does. And this is building up to the Riemann hypothesis which of course comes from him. All right. So this is now called the Riemann Zeta function. All right. The Riemann Zeta function we would say on the left hand side is the analytic continuation of the generalization of the harmonic series. Okay. Riemann Zeta function is a lot easier. All right. Right. Because the generalization of the harmonic series only converges when the real part of S is greater than one. And what Riemann wanted to do with this is he is interested he's interested in the gold too. He's looking for prime numbers. He wants to know everything about the prime numbers. This he sees this. All right. But what more can we do with this? There's something really important that he wants to do. And it's kind of a bold attempt. But he pulls it off. And what he wants he's looking at this and he said I wonder if this is going to tell us anything about the prime counting function. So the prime counting function if you are unaware is simply you take the number of primes up to a given argument count them up and there's your output. So for instance if our input is the number six and then we count the number of primes we got what how many two is prime three is a prime five is a prime. Okay that's three primes that are add up to six or that not up to that we have the number of primes up to six. And so the output is three. And we could do this for any natural number. Well it becomes very cumbersome when you get two very large inputs right. It's not a it's not a function that's easy to work with. Just plug into a calculator. So if we're talking about prime numbers up into you know large large hundreds of digits like some of those used in encryption algorithms it doesn't it's just a practical to take the same amount of time to calculate as it would from the time of Eratosthenes up to Riemann. Okay so he wanted to come up with a function where he could just plug in a number x and it gives x some value and outputs the number of primes up to x. Okay and he wrote a paper it was called on the number of primes less than a given magnitude. And as he was on the way to this one of the things he needed to do is analytically continue the generalization of the harmonic series. So he came up with a functional equation. And so now he can just plug in so functional equation he wants for the zeta function. And one he wants a he wants a functional equation for the number of primes less than a given magnitude. So this is the big thing that's kind of we're going to focus on as we go forward. And I definitely will show his results because it's very important in terms of prime numbers and it's very important in terms of the Riemann hypothesis. All right but this function comes along this is what he gets on the way it's his tool it's called the functional equation of the Riemann zeta function. And it's a product of five function into itself. You see we got two to the power of s that's a function pi to the power of s minus one you know what pi is you know the number that's ratio between the circumference of the circle to half this radius or two times this radius and sine of pi times s over two. So then with the sine function I'm assuming I hope if not you can look it up google it gamma function of one minus s. I'm going to touch a little on the gamma function because it has some interesting results that lead up to the Riemann hypothesis. And again we then have zeta of one minus s. All right so we got zeta of some argument on both sides. This relationship between s and one minus s is very important in terms of the Riemann zeta function. But this is his functional equation that he is able to get his final result in the paper on the number of primes less than a given magnitude. And we're going to take take care of that and go over that in the next video. Thank you this is just the introduction. I understand if you are an expert at number theory and the Riemann hypothesis all of this is review. But I want everyone to be on the same page when we get to the end of part four. Thank you and see you in the next video.