The Riemann Hypothesis: How to make $1 Million Without Getting Out of Bed

I'm going to tell you a little bit about the so called Riemann hypothesis and the subtitle of the talk is how to make million bucks without getting out of it so let me explain why that's the subtitle it's going reverse order so why don't you need to get outta bed to study this problem in the other sciences physics chemistry biology all you need to do is observe the physical world the real world that we live in I need to say you know whatever it is that you're studying you will want to make your objects observe the world and make a mathematical model a sort of let's boil everything down to the bare essentials so it's a very crude approximation what's really going on you solve the model and once you do that you hope that it's going to predict real-life behavior so here's a very important example of physics in action you need to make sure that the Angry Bird will not get paying over and then tumble the whole castle okay so this is how physics would approach the problem on mathematics on the other hand of course you're influenced by the outside world but you don't need anything other than your mind to think about numbers and concepts coming out of mathematics we're just the same halfway around the world in India or China or on the moon as far as we know any two plus two is still four when you get to the Moon or Mars or in your bed so that's the reason that you can do this you can think about this problem without going anywhere of course it helps maybe to go to finish high school college grad school and so on one of the drawbacks is if we're just playing games in our mind we have to completely convince ourselves as we know what's going on with Angry Birds if you tell me shoot at 45-degree angles and I swear it's going to hit the pig well all I have to do is actually shooting and then we know what it was right and in that we're going to have a party approved in other word for proof is arguments so you have to have a completely sound explanation for why you think what's going on is really what's going on all right let me tell you about this the second part of the sentence million dollar prize so in the year nineteen hundred to one hundred and twelve years ago there was a big famous meeting international meeting mathematicians from all over the world came together and one of the most famous Indian mathematicians the guy needed David Hilbert there is with a hat and what he suggested is he proposed fuse the problems that were around at least some of them but he sort of collected them into here are the most important problems there are 23 events 23 years and most important problems for mathematics that tackle in the next hundred years here are the problems for the 20th century and number eight on his list our Riemann hypothesis fast forward a hundred years the year 2000 not so long ago some of you were born men by then there's a new clay mask Institute that was just founded a year before and it was based at Harvard now a photographer and they posed not 23 but seven problems seven problems for the Millennium the Millennium problems prize problems not only that they posed them again these are problems that were well-known they didn't close them they just collected them and said these are the most important they put a million dollar bounty on each problem you solve even one of these problems they will give you a million dollars in cash and number four on their list is the Riemann hypothesis in fact it's the only problem which appears both on the most wanted list in 1900 and the most wanted list in 2000 so this thing is really one of the most important problems in all of mathematics in case you get so discouraged ah there's not a chance you know there's no way I'm going to be able to solve this thing people have been thinking about it for 150 years number three on this list on the claim at least a two million dollar list is the so-called Padre conjecture which we will not discuss today you may have heard you may watch Matt news probably you don't three years after this problem made its most-wanted list the millennium most wanted list a guy named Herbert Pearlman solved the Padre conjecture so it's not impossible for these absolutely unbelievable problems to get solved I should say he had been working on it much longer than the three years between when it became a prize problem and when he solved it and it's based on 100 years of worth of mathematics and I should also tell you that when they gave him the million dollars he said thanks but no thanks and that's him kicking the million bucks away the proof was enough for him just the satisfaction of having proved this thing was all he wanted so the champion is okay this problems hard if you solve it fantastic you'll be world famous but if there's a tradition of not taking a million you might have some freshman laughing go take it anyway million bucks why not all right so what is the Riemann hypothesis well it was stated in 1859 over 150 years ago Buckhorn I'm reading there is again and it's seen hundreds if not thousands of false proof people that write papers even publish papers circulate papers professional mathematicians amateurs experts in the field have given false arguments arguments then when they show someone else they said wait a second how do you get from this line to this line a miracle occurs well that's not an explanation so there it's seen hundreds of false proofs and whatever it is I'll try to explain what it is numerically it's been verified a billion times it's really predicting what's supposed to happen we just don't know why okay so let's not beat around the bush what is what is the Riemann hypothesis does it's complicated if it wasn't complicated it wouldn't be such a high problem but I'll give you the one sentence version it's that Riemann was the first human being to hear the music of the primes and my hope is today you will - all right so the Riemann hypothesis is a problem without prime numbers so let's talk for a second about prime numbers just to get a warm-up what are the divisors of six what are the whole numbers that evenly divides it please search one two and three our devices will also count six so six is gets on any number divides itself evenly any whole number that divides evenly is called the divisor thank you very much that's a perfect answer one two three and six are the divisors of six now what is a prime number yes perfect and so it has exactly two devices because it can only be divided by one and it can be divided by itself and no other no yes sir excellent question at the time the Riemann hypothesis was posed one was no longer considered a prime number but for a long long time before then one was considered a prime number so today for very many reasons we do not consider one a prime number so one does not count and that's exactly why I say exactly two divisors one is only one divisor it's up okay excellent question so you probably have seen the primes here's a short list of some of them 2 3 5 7 11 you can read I'm sorry - is not a prime - gives the oddest proc absolutely absolutely yeah - is a that's yeah absolutely - is the weirdest of all the primes it's the only even one it is a lot of very strange properties but but it is a pride whereas one we will not consider a prime at least today why are these things important why do we care about prime numbers at all what makes them any different from six the primes are the building blocks of all numbers if you take any number you can decompose it into primes if it's already prime and in that way they are like the elementary particles and physics the in decomposable pieces of matter and our universe that make up everything that we know so any number that we know of can be decomposed into primes and that's why the crimes are so important to the fundamental building book so you can think of - is an electron trees the lepton or photon 101 dalmatians for some reason or expose on if you know when these things are okay now there are in the standard model there are twelve fundamental particles and that can change from time to time so this is another reason why mathematics is different from Sciences that observe the world Newton knew exactly how physics works until Einstein changed it and now Einstein knew exactly how physics works until quantum mechanics changed it and and so on so these things change with time crimes except for one one changes whether we consider one a prime or not change history time but that's just a definition that's just a work the facts about numbers do not change throughout absolutely University so there are twelve fundamental particles how many Prime's are they how many people think there are twenty condos raising more than twenty that's right you should be upset how about thirty or a hundred or a thousand or a million how many cries you think there let me pick up some we all we all know the answer right infinite right there have to be right because we can make a computer program and just keep looking for them as far out as we want that's not a crew that's an observation that's not a proof you know rigorous argument we need a very when you vary it's not something that you just oh it has to be you can just throw your hands up in the air that's not a mathematical proof so today we'll prove a theorem our first airing of the day it are in fact infinitely many prime numbers the proof that we're going to present there are many many proofs of this fact but the earliest proof that we know of is by an ancient Greek mathematician named you click lived in Alexandria modern Egypt and it was given two thousand three hundred years ago pretty amazing the first thing we need to know to present Euclid's proof is what does it need not to be prime so a number is composite if it's not prime what does it mean for a number to be composite how do we tell that is composite well like you said we just checked it some other number divides it and it's enough to check it some prime divides it even so a number is evenly divisible if it leaves a remainder is zero so here's we're going to check it seventy-five is a prime or not well we try to divide it by two that leaves a remainder of one so that doesn't that doesn't tell you anything but if you divided by three it leaves a remainder of zero and so the number has a remainder zero when divided by some prime then it's not crying then it's composite right that's all we need to know for this group so how did the proof go it goes by contradiction let's say I'm wrong let's say we're all wrong and there are only finitely many Prime's okay if we can find a contradiction to this assumption we'll just follow very logically from this assumption and come to something that just doesn't make any sense then this assumption must be wrong and there have to be implemented crimes so this is what's called a proof by contradiction so let's assume for now suspend your disbelief let's assume that there are only finitely many prime numbers let's list all of them there's only five we made 2 3 5 7 11 13 to vote on and on 100 million a billion of Julian a googolplex however many there are there's some largest number which is prime which is capital P so now we have this list of every prime number in the universe and it's a finite list because we assumed it is take this number capital n where you multiply all the primes by each other so 2 times 3 times 5 times 7 all the way up to this hypothetical largest prime ever capital P you multiply all these together it's a finite product of finite numbers if some is just someone integer take this integer and add one too okay so n is the product of all prime numbers with one more attack on the end so let's test whether or not and can be prime well or or let's test if it can be composite so if you take N and you divide it by two well this piece is divisible by two so what remainder is at least one if you take any divided by three it leaves a remainder of one if you take n divided by 57 not prime 59 it leaves remainder one right so any prime number since all the prime numbers are here when you divide n by any prime number at all it leaves a remainder of one and one last I checked is not zero and so this number can't be composite because if it lifts a remainder of zero when divided by some prime number that's what it would mean to be composite so that means in there's big number end of your account is not a composite number but what does that mean by definition it's not composites and it's and it's a crime right what's the problem with that anybody else already picked on you yes back then perfect P is the highest prime number and yet we just show that there's another number which is crying which is bigger than P so n is not this number n is not in our list of all prime numbers P is supposed to be the largest and that's a contradiction and now we need to write these three funny letters quitter at them in Stratham therefore we have proved what we set up anybody so if you observe we didn't actually sit at a computer and write down even a single prime number for this proof it's a proof by pure thumb it's just an argument it's just a discussion between you and me I think everyone in this room is now convinced it can't be the case that they're finding too many prime numbers to have to be okay any questions all right well once you ask about the friends themselves you might be interested in patterns of prize mathematicians life patterns and like finding pattern so anybody know what a twin crime is yes right there it's two primes that are two away from each other numbers P and P plus two which are both prime numbers so let's just look at that list at table that we had so three and five give four by two and they're both prime numbers five and seven differ by 2 11 and 13 different by two 41 and 43 different by two and this is the sequence of the first few pairs of three prime numbers what's the most natural question we would ask when we see a list like this how many how many right are there 10 30 a million are their implementing twin primes again you write a computer program and ask it to give you lists it gives you lists for as long as you like but that's not a proof that's not a rigorous argument that tells you what the answer has to be and always will be forever so what's the answer anybody know you're right that's the correct answer the correct answer is nobody knows 2300 years after Euclid's proof that they're implementing crimes the very next most natural question you could ask patterns of twin crimes we still do not know the answer we don't know the answer rigorously we think we know the answer we think it's yes why shouldn't there be but that's not a proof mathematics requires nothing but absolute proof is that clear all right and no one has given one for this problem here's a little homework assignment for you why can't you use the argument we just gave we just gave Euclid's argument that there have to be many primes try to adapt that argument to this problem what goes wrong think about it you don't have to give me an answer now that's a home all right here's another pattern of crimes that people like these are called marsan crimes anybody know anybody heard of Marsan crimes but you know what a Marsan crime is okay okay these also make the math news sometimes from again probably not here Sunday morning reading it's named after this guy very marsan he was studying them in the late 15th early mid 1600s a Mersenne prime is a prime number which is almost a perfect power of two so the power of the two one two four eight 16 32 and so on and we're set prime is a prime number which is just one less than a power of two so let's just see 2 to the 2 to the 1 is 2 and if we subtract one off of that we get 1 in more sense time one was a private for us 1 is not a friend how about 2 squared 4 minus 1 3 3 is prime that's a Mersenne prime that's our first percent prime 2 to the third is 8 minus 1 7 is prime 2 to the 4 16 minus 1 15 is 3 times 5 now from ok we can keep playing this game forever 32 minus one is 31 this list is much more sparse than the last one but here's a number anybody know what suited of 4 I can be 43 million 100 and whatever minus 1 I'm not going to write it out if it's 10 million digits long but this number is the largest marsan crime we know of today question oh no justice justice character sketches are a lot of two of course what's the natural question are there incrementing what do you think we know about this nothing okay let me tell you a little anecdote there's a very famous lecture by a guy named Frank Nelson Cole there he is there in 1903 110 years ago what he did is imagine if I did this to you he spent an hour his entire hour lecture he spent computing in utter silence on the blackboard just computing what was he computing something very complicated no well that's something a little complicated but it's nothing that you can do on your own with thousand hours of paper and pen if you're tooted to ^ 67 well subtracting one even I can do but this is 2 to the 67 with an eighth day that's a huge number and a huge computation it took them about a half hour to get all of the numbers down and do the long you know you have to multiply down in a clever way huh just the most trivial way and he gets that number that takes a half hour on the other half of the blackboard he writes down that number 103 million and so on and you write sounds that number 700 billion and he stands there and in front of a room full room like this of professional mathematicians not a theater it multiplies this out and as he's nearing and you're hearing losing us as people realize that he has factored to to the 67 minus one which is not a Mersenne prime because it's not a primer okay so okay on the computer today you do this in a second but I hope you appreciate the magnitude of difficulty of this problem and that we really don't know very much about Mersenne primes and much in the same way when you get to very large prime numbers twin primes are good so let's get back to the Riemann hypothesis does the problem we're trying to understand today what's the moral of the examples we just discussed prime numbers are very mysterious and we know very little about them now the Riemann hypothesis says two things about it number one it says that the prime numbers are beautifully structured they have an amazing structure and number two is that prime numbers are completely random if it sounds like those two things are in direct contradiction with each other you'll see what I mean so I have to explain exactly what these statements are so let's start with number two let me try to explain to you why Prime's a random okay here's a guy maybe you've heard of him Mobius he's named after his strip the other way and what he did is he gave a spin to numbers so one has spin up it's one it's got an F spin out two has spin down three has spin down what do you think for the spin of four is how many people say up how many people say down well it's as no spin you didn't know there was an option seat how about five spin up or down down how about six oh how about seven down eight go spin nine no spin 10 is out here the next 10 what's the pattern yeah what's the connection alright let me pick on you again what about 9 12 is not a square it has no spin it's an excellent guess oh yeah oh yes so there are four rules rule one is that one has spin up rule 2 is that any prime number has been down what's really it's almost what you said whatever a number has a square divisor whenever there are two prime factors of the same prime in the number like 4 which has some twos in it not 8 and 9 have are divisible by squares 12 is divisible by 4 which is a square those spins cancel out and that number has no spin and what about a number that's a product of distinct prime factors well each prime factor contributes a down spin and 2 down students cancel out coming up spit ok so let's just see if we understood with that what that means let's compute the spin of 30 so what do we need to know to compute a spin of something we need its prime factorization so what's the prime factorization what's the prime factorization of 30 pretend you're deaf yeah 2:35 good so what's its pink to gives it a spin down three gives it a spin down five gives it a spin down three down spins is the same as one down spin in just the two two down spins gets loud so 30 has spin 10 okay that's Mobius funny now it's nicer to think of instead of arrows numbers so let's just convert an up arrow into a plus one a down arrow to minus one and no spin is Europe okay so here's mo business function again if some squared divides the number n then you get 0 if n is a product of an odd number of distinct crimes then there's one spin that's not matched up and if n is a product of an even number of prime factorize then it has spin plus 1 just as we just say so let me just put the table there again so here's the first line is just the numbers the second line is their spin as we just saw in the third line is the möbius function which converts arrows to numbers any questions now what are the partial sums of the möbius function so what do I mean by partial sums you pick some number X and you add up the first X spin let me explain with that name so let's start with X at most 10 so we start here at 0 when we get to 1 1 head spin plus 1 so we take a step up and when we hit the 2 to it spin down it has Moebius minus 1 so we go down what happens 3 go now what happens at for nothing we just stay where we are but m25 down six up seven down eight and nine we don't do anything can we go up you see what's going on let's go out to 20 so the first 10 is exactly what we just saw and then 11 goes down 12 doesn't do anyth 13 goes down 14 goes up and so on okay you do it out to 100 and you do it out to a thousand I can do it out to 20,000 isn't that weird yes it is weird thank you this is what we mean by Prime's are random now I have to explain what that means so let me explain what I mean by a number being random well what does it need to be random say I have a three sided coin I don't know how you get a three sided coin but suppose it flips with probability of 13 lands up probably 230 lands down probably a third atlantes I don't know on its side or something if we were to add up these partial spins out to 20,000 here's one sample picture now in probability theory there's a very famous theorem called the central limit theorem I'm not going to prove before you today but basically what it says is you should compare a curve like this to the graph y equals root X plus your mind so there's this it's a parabola Y squared equals x so it's just the parabola but it's on its side and what you should be seeing what this theorem says is this hat should stay inside the curve it's allowed to touch the curve it's even allowed to go a little bit outside the curve because it's very formal mathematical that sounds like I wish you watching James before mathematical statement which I own it tell you because it's complicated but at the essence of what it says you should compare this curve to the curve brunet and it should stay inside this bag let's see what's going on with Mobius monkey so let me put these Mobius tones that we saw there it is can you even tell which path is which anymore yes some yes some no it's hard to tell the Riemann hypothesis says you can't tell it looks random all right let's just test this out anybody got a coin raise your hand if you have access to something you can flip it there are three of you with coins in here ask your mom okay raise your hand if ever anybody who's got a point if you have more than one coin you can give it to your game we're gonna play I hope they give it back to you okay yeah points whoever has a coin keep your hand up now we're talking all right here's what we're going to do we're all going to flip our point okay you flip you catch and you see what you have all right everybody flip now once you know what you have either heads they just hold your hand up in the air you're going to tell me one at a time what you have we're going to go around Rome you're gonna tell me what you have okay ready wait a second don't tell me it not listening okay I had a head who's got a hand up please Hey keep going as fast as you can heads tails kale kale and so just keep it up just going down hey hey hey hey I heard that I hear the tail head to the tail tail tail tail tail good tail tail and that may be a mistake it no you're going tail tail head head tail head hey hey tail-head mic window two in the back what do you got hey over there tail alright that's that's your beam up let's see if this works that's our graph this is the sequence of heads and tails that you just yelled out at me and as you see look at it we had about 40 people call things out if every single one of you had a head this crap would just go straight up to 40 the square root of 40 is like 36 is the square root of 36 is about 6 and this is 6 and here's give or take 6 and look at our rep obviously I couldn't break this that's what the central limit theorem says and what the Riemann hypothesis says is that the sums of the möbius function behave in exactly the same let's go back let me try to explain to you a little better ok here we go alright let's look at another version of it there's another reversion well for that we need to remember log so what's log not that it's the logarithm function y equals log of X so if you don't know what this is you don't really need to know if the inverse function is e to the X 1 you don't need know what that is let me just remind you that E is a number between 2 & 3 it's called II after Euler although it was known long before him here's a graph of 2 to the X and a graph of 3 to the X and E is between 2 & 3 so there's the curve that sits inside and what does it mean to be the inverse it means if we draw a to the X and we flip everything along the line y it was exit place the role of x and y we get this curve y equals logging okay if you haven't seen this before it doesn't really matter it's just some function gives you some number its value at 1 is 0 and there's another property of it W so let me tell you what the fun Mangold funky I looked all over Google Images I couldn't find a picture of on mango so here's what it gives you for the number one it gives the value 0 but the number 2 it gives the value log of 2 which is like 0.69 for number 3 is the value log of 3 what do you think is in all your fur for log 7 is it interesting yes log 4 is a better guest it's log of 2 how about 5 5 okay how about 6 zero you do I'm up Oh clicked you soon okay what's it gonna be log it - good how about 9 rocket 3 you guys how about 10 zero so just I'm just writing out the numbers so you see log of 3 is just 1 point 1 log of 7 is 1 point 9 what's the pattern yes sir close it's sort of what you said you get log of a number if the number is an exact power of a prime so for example 8 is 2 cubed so it's a power of the prime - and that's why you get a log of 2 + 9 is 3 squares and that's why I see a log of 3 and 5 is 5 to the 1 which is why you see log of up and what about numbers that aren't powers of crimes just put a 0 6 there's about two different crimes 0 n is 2 times 5 you put it here ok let's look at the partial sums of this long mangle function here they are again let's let's graph them out to 10 so what does this say we start at the origin we get to 1 we add 0 so we don't do anything we get the 2 we add point 6 9 we get the 3 we had one point one we had before we add 6v is a 5 we add 5 we get mistakes we had nothing we don't move and so on ok so if this little step function like that that's the other sums of the home angle function how's the 20 it looks like that so again the first half is what we just saw out to a hundred it looks like that and out to a thousand it looks like that what does that look like it looks just like the graph of y equals x let me show you why equals x that weird let's see you see it's sort of wiggling around the line y equals x let's see what happens if you subtract it off y equals x from that stuff so what do I mean by that we're going to take these partial sums and subtract X Aquajet so let's go out to 10 again here's the graph out the 10 that we just discussed here's the line y equals x how are we going to take at any point we're going to take this number and subtract that number so we get first at least we'll get negative numbers so there's the difference that's what that's the wiggle it working here's that same wiggle so you see this little pattern out the 10 we're going to see that same wiggle pattern out to 10 here but then we'll keep going out to a hundred there's the wiggle pattern and out to a thousand there's a wiggle pageant what does that look random he looks random how do we test if it if it looks random identical extreme yeah let's test it against this problem there is the problem well you don't see it yet but when you go out quite quite a bit further it's going to start hitting the boundaries on the problem but it's supposed to stay more or less inside supposed to behave just like a random function okay so let's recap the prizes are random in two ways we can look at partial sums on the möbius function that looks random we can look at partial sums of the bomb angle function with this X subtract it up that looks random is there a relationship between these two for now let me try to explain that running low on time so remember long ago one was zero and the only other property we need is if you multiply two numbers and take their logs the same as adding the values of the law here are the values of the von Mangal function which come out as fings I'm just reminding you here are the values of the sorry the möbius function here the values of the bond angle function and there you know log of 2 is just some number points let's go back to that our favorite number six over its divisors one two three and six let's point by manual function of six on one side and on the other side let's take the möbius function and multiply it by log and evaluate that as the divisor so we'll put Mobius of 1 log of one Mobius is 2 log of 2 Mobius if you log of 3 and Mobius is just log of 6 and let's add all of these numbers up ok so what's my angle to 6 use your table of six bond angle sorry lambda of 6 is zero okay zero how about Mobius of 1 Oh B sub 1 is +1 but log of 1 is 0 Mobius if 2 is Mobius so whole piece is known within you and you don't know your Greek letter o case is the top guy and okay so Mobius gives you a minus 1 that block to Mobius of 3 is minus 1 and plug 3 and Mobius of 6 is plus 1 times log of 6 so let's just check here is 0 and here zero somebody's already got it well I have 6 by that formula is log of 2 x plus log of 3 here's a minus log of 2 and a minus log of 3 and what do you get 0 they all cancel out so here's another homework proof proof give a rigorous argument get in complete generality if you take a number in and you write all of its divisors just like we did D 1 D 2 up to D K and you take the bottom angle function and you take Mobius at the divided kind of log of the divisor you add this up over all the divisors you will always get to fix it so this is the relationship between the bond manual function and the Mobius function they're very intimately related to one another through this function log ok let's recap again the primes are supposed to be random if you look at partial sums of Mobius or if I'm angle minus X they both stay inside this curve root X and that's what it means to be random that is the Riemann house during my process says primes are random Buttrey month as I explained saw something much much deeper and it's that primes are perfectly structured in fact he heard the music of the price so that's what I'm going to try to explain ok so we have to talk a little bit about music and for that let me introduce you to a few friends of mine this is best rose and May so this is best this is an alto saxophone this is not bestest best day I don't know if you guys can see rundas rubber bands all over it because one of the hosts came off this morning let's hope it's the work that's best this is rose say hi rose anybody know what that's called sopran outside right rose see she's up and running this is rose and May anybody know what me is clarinet right music fine sitting around for a little while okay so we have these three horns an alto sax a soprano sax and a clarinet there so the saxophones were another bag invested by a dog Jackson mid 50s and I want to present you with three mysteries about these swamps so mystery one let's measure the length of the soprano in the clarinet so if you see well we have to unwind this is the soprano is a little wrapped up but if we unwind it you see they're about the same length they're almost exactly the same length these too long this one's one doesn't start to see that length is about 26 inches you can even tell here at 26 27 inches okay how about the alto the alto is if you were to unwind it it's 39 to 40 inches and then you would notice the relationship between 39 and 26 what's the ratio from one to the other two thirds right or three yet twenty-six times three halves is so we can see that by comparing the clarinet to the sex today also and seeing that what's left over is about half of the clarinet one and a half the length of the clarinet it's the length of the sexual alright why these lengths why is this ratio what's so interesting about three half mysteries too let's study the lowest notes on each horn so on a sax on the alto that's the lowest note on our clarinet that's the lowest note they're pretty close right the lowest note ah there is almost exactly the same note as the lowest note I don't help to help it just has one one more half step lower than a clarinet so the lowest note on may is almost the same as the lowest note on best part as opposed to Rose that lowest note is much higher than the lowest note on the other guys it's in fact three halves I'll try to explain what that means three halves hi so question - why is this happening question three the saxophone sound is often described as being lyrical and voiced light and very sort of warm that's kind of a warm quality to it whereas the clarinet is often described as having a very pure sound it's a very you can tell the timbers are completely different from a saxophone to a clarinet so question three is wider than Tambor's okay so let's summarize our mysteries as far as life goes the alto sax is the odd man out Rosen's the soprano on planet I've got the same length but if you look at lowest note the soprano sax is the odd man out Sparrow has a much higher lowest note of any alto for the clarinet and as far as hammers concerned the two saxes have a warm sound and the clarinet has a pure sound clarinets geography so what are these things happens well to understand it we have to understand what is sound so I'm running very low on time let me try to rush through this sound is a mosh pit you have a whole bunch of people in a room they're all dancing and if one of them is we shove and one guy shows just shoves in a random direction everybody nearby starts to fall and tumble and if you're Lenny of this from overhead you see regions where people are squished together in regions where there is little space so sound is a mosh pit of air molecules what's really going on is as my voice tries to communicate with you I'm squeezing air and that air gets pulse towards your ear and you have an eardrum on one side is just normal pressure inside your head on the other side is normal pressure in the room that gives silence but when there is a disturbance when the pressure is higher your eardrum pushes in when the pressure is lower your eardrum pushes out because there's pressure on the others so that fluctuation of the eardrum is exactly what we hear as sound so when you see a graph it's a graph of the pressure it's not that sound is going up and down sound is being pushed towards you and cold cold air is being pushed towards employed okay so if you see people have a misunderstanding of it there's nothing going up and down what's going up and down is the measure of the pressure okay um so again what what do we hear let me explain what that is supposed to represent you get it so for a while there was no sound all of a sudden there was a big change in pressure and you're here and then there is no change in pressure again so that's one pump now if we were to clap once per second so that's called a 100 person Hertz it would look like this join me don't rush yeah so that's one percent the time between each clap between each fluctuation is called if the period one over the period is the frequency how many class per second and as we already said that's one first now what if we were two claps off what should the graph look like yeah it's exactly the same difference in time but the amplitude is changing cinema big amplitude a big change in pressure it's a small change in pressure we hear that is volume so amplitude is volume gets measured in decibels which is 1/10 of an Alexander Graham Bell that's about all right what if we double Prime okay that's 2 Hertz two beats per second and the graph just has more wiggles let's try this what if on one hand you can do this on your knees maybe on one hand you clap regular time on the other hand you clap double time get it what is the graph of that look like there's it's big beep and then a little beep right when both hands clap you got a big beat and then when only one hand got to get a little weak that is exactly the sum of this picture and this picture you see if this goes up a high one and this also goes up a high one then the sum of the two go up a height of two and here is one plus zero point one okay so everybody seen it so when you add two sounds on top of each other that's the superposition of sound waves just add how about one two three yeah not that how about two to three not bad double you're getting it how about 43 stop somebody getting it let me give you a little harder one four to seven alright let's recap what can we experience there Beatson is what we experience what we hear is frequency is how many beats a second and in platoon is how strong is each fluctuation hell out so amplitude is volume so we were to sing this what would it sound like let's try to destroy all together so we don't change the notes the period is still the same we just think quietly that loud in five days yeah okay that's amplitude as a way how about frequency frequency we hear is pitch a fast frequency over here is a high pitch and a low frequency over here a slow wave yours alone hi-hi no we here is a as a fast frequency so let's try to sing this ready yeah so frequency is pitch and amplitude is volume if we speed up that little class that we were doing not twice per second not three times not 100 times 440 times the second what we would hear is that's an a 440 the reed is fluctuating 440 times a second and then fluctuation is making its way to you your brain interprets it as sound that's an a what if we were doubling the forwarding to double that frequency not from 400 to 800 e you would hear just one octave puckered okay so changes in frequency corresponds to changes in the note what about Tambor what about quality of sound how do we understand that yes someone somebody said okay you're fine if I play the same amplitude and the same volume same amplitude on both those they still sound different same frequency same amplitude what's different is the shape the quality the sound is a turn by wave shape so you can have a wave it looks like these triangles these are feeds so you can have a square wave or one of these sinusoidal waves they sound different even if they have the same frequency in hand and how do you understand this through calculus which I not going to go into but it's worked out at Joseph Fourier who showed that any way whatsoever can actually be decomposed as a superposition just as we saw before a very simple pure wave so you just have to use the right frequencies and amplitudes let me let me try to explain so let's decompose this two on one side and one on the other side let's decompose that beat into pure wave so these are waves are going around at 1 Hertz this is a sine wave barely fluctuating at all very small amplitude here's a tumor that's wiggling twice as fast there's a 3 Hertz and 4 Hertz so you see the amplitudes are different and the frequencies are integer multiples of the original frequencies so this is wiggling four beats a second and so on there's the tenth contribution and if we superimpose the ways we just add them on top of each other when we go up to five Hertz this is what the superposition looks like I don't know if you can tell the heartbeat underneath the superposition and if we go out to ten Hertz that's fourier spear the theorem is you just make these very simple waves and you add them up and you can get any way you want and that is why you can play rihanna on your ipod is because all the computer can do computers can't store complicated waves like they can store numbers all the computer has to do is store the frequency and the amplitude of a bunch of numbers and then it can make any shape of wave whatsoever so let's solve our three mysteries here they are again saxophone is the odd man out the alto is the odd man out lengthwise as far as the low note the Sopranos the odd man out tambour the clarinet is the weirdo so let's answer the camera question the sound again is determined by the resident waves the clarinet is a cylinder the length of the circle here is almost exactly the same as the length here it just looks like a strange cylinder across everybody see that whereas a saxophone looks like a cone it starts out very very small and gradually it's bigger never minded twists gradually gets bigger and bigger so it looks like a cone and so the fluctuations the fact that this declarant has a cylindrical bore means it's overtone frequency doesn't have even multiples if I over bloah you're supposed to be if I play all of the multiples of the original way so that's the actual full overtone series now if I don't change my fingers at all and I just put more energy it's like jumping from an s valence to T valence electron you'll hear it's skipping everybody note that it should be play that's what you should be here is instead what you're hearing and that is just by changing how much pressure it but how much power it book through the point a saxophone has a conical bore as we just said and that allows you to overload all harmonics now the winning isn't a simple sine wave it's a little more complicated section sine x over X but it allows all frequencies to overblow and you should hear okay okay you heard almost all the harmonics I'm sorry she's having a bad day but here's the point is you're hearing the full overtone series on a sax which you don't hear on the clarinet so the answer to question two why is the lowest note different it's the same answer it's the physics it's the mathematics of what notes are the resonant waves going into four eh theory and it's just the fact from calculus that if you have a cylindrical bore a bore that looks like a cylinder air resonates at half at 3/4 the length your resonates the same frequency as a cone which has three times as long so three and three halves times long so even though the alto is quite a bit longer than the clarinet they have the same lowest note whereas the soprano as expected has a higher lowest note because it's shorter than the alto how about question one same answer we just we just gave him the lengths are different because that's these are the frequencies going into the one so all of the mysteries are solved by understanding the resonant weight the resonant frequency math explains everything so let me give you three homework problems here's some some things to explain what's going on if we pop it up what's going on with that top here's another phenomenon thunders game how about won't be funny what's happening there that was weird right and the third one what happens we've only been discussing single notes but of course music is about not just single notes music is about harmony notes playing together what's harmony but several voices playing at once what's going on in you here there let me give you a hint remember when we were clapping two two three goodbye alright let's get back to the Riemann hypothesis what what was the Riemann hypothesis what Riemann actually proved there's a mathematical theorem that you prove and that theorem is basically the analogue authorities theorem you discovered some amazing wave over time just four more minutes of your time if that's okay these are some of the ways that we month is covered and if you look at their superposition if you just add all these ways up you get something that looks like this does that look familiar it's the we already saw this curve it was the way that we got by subtracting all s from the sum of the bomb angle let's go back let's go back to the very beginning from the very top to recall the bomb angle function it has these log values and it tells us which numbers are prime or prime powers but there are very few prime powers so the prime numbers are the ones that are contributing the most to that and we looked at its partial sums and we saw a picture like this and we immediately so the jumps tell you where you find products on friends house and we immediately observe that there is this the graph just looks like Y to the x y y equals x right that we saw with our eyes there it is is the fundamental frequency of the trend it's the main frequency it's the one that tells you what the harming what the what note the primes are playing but to get the full characteristics it's the one that sounds the loudest you get the full characteristic you need the superposition of all of harmonic waves that's Riemann sphere so let's just have this first wave remember this wave one that I showed you rimas weight one let's add it on top everybody see what happened let's go back we were there that's why it was X we add it on top it's better approximating that step function let's add a second wave here's way too on top of already added wave one there it is how about going out to wave tank you get that how about going out to wave a hundred Wow you see it's getting closer and closer to exactly matching that step clock so that's what Riemann proved that there are these beautiful pure waves just like if you're open for Ian's case they're determined by two numbers anyways key numbers once you numbers putting the specify where you can see an amplitude how fast a wiggle and how loud to play and what Riemann did is plotted the frequencies in the amplitude wave zero that fundamental that y equals x line has frequency zero it's not wiggling at all and amplitude one it's loud it's it's sounding the loudest of all all of the other ways they have some frequencies which we don't know care you can compute what these numbers are and they have all of their amplitudes playing at exactly volume 1/2 that 1/2 is the same 1/2 as the line y equals x to the 1/2 that we saw in the randoms so what do you think the Riemann hypothesis States looking at this picture what do you watch Yuri conjectures where do you think wave limit is going to be over here over there over there over there yeah they all the Riemann hypothesis says all of the waves that add up play at exactly the same volume there is a perfect harmony this is the perfect structure of the prime numbers all of the waves the music that's making up the primes there's this very clear line y equals x and everything else is contributing much much less and the amount is exchanging by path that's the perfect structure and Riemann sphere alright let me just say I stole these pictures from Wikipedia and Google Images you should also check out this wonderful book by Marcus de Soto I called the music of the prize and eventually I'll throw some of these pictures and computer programs on my websites you can download

Channel: Yale University

Views: 232,506

Keywords: Pathways, Pathways to Science, Science Outreach, Mathematics, Yair Minsky, Minsky, Math Mornings, Math Mornings at Yale, Alex Kontorovich, Riemann, Prime Numbers, theorem, Riemann Hypothesis (Idea), Mathematician (Occupation)

Id: yhtcJPI6AtY

Length: 65min 41sec (3941 seconds)

Published: Thu Oct 31 2013