The Queen of Mathematics - Professor Raymond Flood

this lecture is about one of the core concepts in mathematic in fact whole numbers or integers such as 1 2 3 4 and so on now the study of the properties of numbers is usually called number theory and it's a vast and fascinating field of mathematics it's also sometimes called higher arithmetic and among the whole numbers are integers the primes are particularly important and I'm going to soon define and introduce them it turns out that there's great difficulty in proving even relatively simple results in number theory and this calls the grid German mathematician called Friedrich ice to remark that it is just this which gives the higher arithmetic that magical charm which is needed the favorite science of the greatest mathematicians not to mention its inexhaustible wealth where and it so greatly surpasses other parts of mathematics gose is often known as the prince of mathematics Andy called mathematics the queen of the sciences and considered number theory the queen of mathematics and it's from guys that I took the title of the lecture I will start by introducing and defining the prime numbers I'm giving some examples then we will quickly see their importance as they are the building blocks from which all the other whole numbers are constructed this result is called the fundamental theorem of arithmetic one of the most crucial questions about primes is how many are there and the answer to this question is given by one of the most beautiful proofs in all of mathematics the proof was given in Euclid's elements which featured so prominently and my last talk on geometry the rest of the lecture is in three parts first we will look at some approaches to finding or generating prime numbers then we look at some properties of the primes that show her random and unpredictable they are or seem to be finally we look at the primes in a different way and you are predictable they are and introduce what is called the prime number theorem and in discussing the prime number theorem I will introduce one of the most intriguing and important functions in all of mathematics the Riemann zeta function this function lies at the heart of one of the most intriguing and difficult questions and mathematics the Riemann hypothesis the question is do all the solutions of a certain equation have a particular form and the full statement that I'll come to at the end of the lecture and say more about is do all the non-trivial zeros of the riemann zeta function of real part a half now answering this question could win you a million dollars it was worth coming the million-dollar prize is offered by the clay mathematics Institute now based in Oxford and the solution to the Riemann hypothesis is one of the seven problems that were originally posed one of these seven problems the pawn correct injector has now been solved the clay mathematics Institute proposed the prizes because as they say on their website the prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium to elevate in the consciousness of the general public the fact that mathematics the frontier is still open and abounds and important unsolved problems to emphasize the importance of working towards the solution of the CHEO of the deepest most difficult problems and to recognize achievement in mathematics of historical magnitude if you want to find out more the web address is on the handout available at the end of the lecture and in the lecture I will end by showing you how the Riemann hypothesis is related to prime numbers now to start at the beginning with what is a prime number so a prime number is a whole number greater than one whose factors only factors are itself and one so examples to let you absorb the definition 2 3 5 7 11 13 17 19 or prime but not 9 equals 3 times 3 because of this factor 3 divides into it or 15 equals 3 times 5 because it has factors as well or I would new you would want to know 2013 this years did is not prime 3 times 11 times 61 also next year's theaters now prime because 2 is the only even prime and 2014 is divisible by 2 2015 the year after is divisible by 5 2016 again is not prime but 2017 is going to be a prime here now the primes are the building blocks from which all other whole numbers are constructed this result is so important that it's called the fundamental theorem of arithmetic every whole number can be written as a product of prime numbers in only one way apart from the order in which they are written so 30 is 2 times 3 times 5 it's the product of three prime numbers 2 3 and 5 and no other collection of primes can be multiplied together to give 30 48 is 2 times 2 times 2 times 2 times 3 and just to show you 22 million 12,000 and 13 is 19 times 50 3 times 20 1859 not the surely doesn't cheat when you get get larger numbers and this also gives you one of the reasons why we don't count 1 as a prime because if we took 1 as a prime then this fundamental theorem of arithmetic would not hold because for example you could write 15 as 3 times 5 or 3 times 5 times 1 or 3 times 5 times 1 times 1 etc so it's a it's tidier to exclude one from being a prime so how many prime numbers are there here I've listed the prime numbers up to 100 for up to 10 then another for up to 20 two more in the 20s to in the 30s 3 and the 40s to in the 50s and then up to having only one in the 90s so writing it out like that in case you might think they're going to end and there would be a dearth of primes but it turns out that there are an infinite number of primes and the proof of this result I think is one of the most celebrated and famous in all of mathematics it's also one of the most beautiful now hard could you go about proving it by I mean that there are an infinite number of primes well the most obvious way I suppose would be a constructive way to find some sort of formula that we generate the primes and if not all of them at least enough of them to show that there were an infinite number but it turns out that that's incredibly difficult today and we'll be looking in a few minutes at some of the approaches that we've taken is too high to go by generating prime numbers the way of approaching it is due to Euclid and it is based upon proof by contradiction and Euclid first did it in the elements and book nine proposition 20 and there he proves there an infinite number of prime numbers and he does it by contradiction he assumes that there are only a finite number of primes and well Euclid assumed there were only three I'm going to stretch you a little bit further than that I'm going to assume that there are n Prime's P 1 P 2 P 3 P 4 P 5 P 6 up to P n but whatever it is there are a finite number of them we can write them down because there's a finite number and then we construct this very clever number n which is the product of all the primes now remember there's only finite number of prime number plus 1 the product of all the primes plus 1 this number is not divisible by any prime because P 1 can't go into it because of the clever bit of adding the 1 onto it P 2 does not divide into it P 3 does not divide until P 4 does not divide into it so none of the Prime's P 1 P 2 P 3 of the P n divide into it but remember the fundamental theorem of arithmetic that told me and you all know that I saw it even though it's lights in my eyes up here you all believe that every number is a prime or made up of primes that's what the fundamental theorem of arithmetic says so N is a prime either is a prime not in the list or is made up of primes not in the list in either case we have got a contradiction and our original assumption must be false there cannot be just a finite number of primes there must be an infinite number of primes and let me because of the algebra within less of the way it's represented look at a couple of concrete instances to show you that both of these cases can apply assume that the only Prime's are 2 3 & 5 and let construct the clever number 2 times 3 times 5 plus 1 but we can evaluate it and it gives you 31 and what is 31 it's a prime not on the list assume the only Prime's are 2 3 5 7 11 and 13 and let n again be their product 2 times 3 times 5 times 7 times 11 times 13 add on one and you get 30,000 and 31 you break that up into its factors and you get fifty nine times 509 two primes which are not in the original list so if you assume that there's a finite number of primes you can show a contradiction so we've got an infinite number of primes so we've got no worry that they're going to run right and I'll just show you the well show you the Euclid proposition 20 and book book 9 where Euclid does it for 3 but he doesn't he does it for lengths and his numbers are represented by lengths and his numbers are are measured by primes rather than divided by primes but it's essentially the same argument over two thousand years ago so anyway now that we know that the primes go on forever let us look at the temps to find or generate some of them and possibly the old 10 did sperm around the 3rd century BC it's named after the Greek Aristophanes who also by the way estimated the circumference of the earth and it's called the sieve a very stuff than is so illustrated by finding all the prime numbers up to a hundred what we do is we write down all of the numbers up to a hundred and delete the number one then we put a circle around two and delete leave it but delete all the multiples of two there's the circle around two and then all the multiples of two are deleted before the six the year two the 10 the 12 all even numbers beyond that then you look at the smallest number that's left that's three that also is prime you put a circle around three and delete all the multiples of three we already six had gone but nine I goes right 21 not goes you continue it so restful it's very relaxing the lowest number five for the circle around it eliminate all the multiples of five in the table and then finally you put a circle around seven and delete all the multiples of seven in the table and we have don't have to go any further what we're left with no are all the primes up to a hundred and the reason we don't have to go any further is very nice because if you take for example 91 91 is 7 times 13 so it's a multiple of 7 so it's already been deleted and in fact if you take any number smaller than 100 which breaks up into prime factors one of those prime factors has to be less than 10 they can't both be bigger than 10 how their product would be bigger than 100 so if you've got a number that's less 101 of its prime factors has to be less than 10 so it's already been scored ight because we were scoring I to the multiples of all the numbers less than 10 and in general we only have to score right the multiples of primes up to the square root of the number that we're looking at so 100 the square root of 100 is 10 if we wanted to find all the prime numbers up to 10,000 then we would only have to score right the multiples of the prime up to the square root of 10,000 which is a hundred so it's a relatively effective way of finding the primes it doesn't exhibit them explicitly it gives you a way of being able to find them but is there a more direct way of generating primes listen approach taken by the grid Leonard Euler it was one of the most prolific mathematicians of all times he produced over 800 books and papers in a wide range of areas from such pure topics as number theory and the geometry of a circle biomechanics logarithms infinite series and calculus to such practical concerns as optics astronomy and the stability of ships he also introduced the symbol e for the exponential number F for a function and I for the square root of minus 1 in the words of the French mathematician Laplace Reed Euler reorder he is the master of us all and he came up with this quadratic a very simple expression N squared plus n plus 41 which has the following amazing property that if you evaluate work out what it is at the number when n is equal to 0 so that 0 squared plus 0 plus 41 giving you 41 you get a prime number when you work out what it is with n is equal to 1 1 squared plus 1 plus 41 gives you 43 another prime when n is equal to 2 plus 2 plus 41 that's 6 plus 41 is 47 it's harder to do the adding up when you're standing up here and if you're sitting down there when it's 3 9 squared + 9 + 41 giving you 50 3 up to n equals 39 it generates prime numbers they're not the ordering of the prime numbers it's but it gives prime numbers what doesn't go on forever when N equals 40 it gives you 16 81 when you evaluate it and that's not a prime its 41 squared and you can see why this approach is not going to work for everything from the next one when n is equal to 41 because then you've got 41 squared plus 41 plus 41 every factor is divisible by 41 so the whole thing is another attempt at generating primes was suggested be firma pierre de firma he spent most of his life and Toulouse as a lawyer he considered mathematics as a hobby he published little and communicated with other scientists by letter firma was the first important European number theorist since Greek times and he resurrected the subject with stunning results these other many areas of interest many area of interest was analytical geometry which he helped to find and Furman conjectured that of Anna's an integer then this more complicated expression 2 to the 2 to the n plus 1 is prime night to work this out one starts with the 2 to the N and evaluates that and then works out to to that par so for example when n is equal to 1 we work out what 2 to the power of 1 is that's 2 so then we work out 2 to the 2 which is 4 plus 1 giving you 5 when n is equal to 2 you work out the - squirt giving you 4 2 to the 4 is 16 + 1 is 17 and this is one way you can tell the difference 2 to the 3 we work out to to the three first of all which is it to ^ viet is 256 i'm adding on one gives you 257 on each of those three is prime as is the fourth one which is 65 thousand five hundred and thirty seven but firm i was mistaken and assuming that his formula worked for subsequent values of n these numbers get very big very quickly because when you put n equal to five the number you get is 2 to the power of 32 plus 1/2 to the power of 5 is 32 2 to the power of 32 and that is that number which is four thousand two hundred seventy four million i think it may be over four billion if we're taking a billion as a thousand two million um that's divisible by 641 remember and vermis days there were no computers around and what this was shown by euler again over a century later and no other firma prime no other Furman number has been shown to be prime so it worked for the first four but then not very successful after that my third example of generating primes is due to the Marin Mersenne a minimized friar who lived near Paris in the early 17th century he believed the scientific discovery should be made available to all with the ZN we carried out an extensive correspondence with most of the leading european scientists of his day acting as a clearinghouse for new scientific results he also regularly initiated regular meetings in paris which mathematicians could meet to discuss their latest findings and these meetings layered in 1666 to the finding of the French Academy of Sciences by Lu the fourteenth no I am er CN prime is a prime of the form a power of two minus one so we're looking for Prime's whose form is a power of two minus one so let's run through some of them if we let n equal to 2 we get to 2 minus 1 2 to the power 2 minus 1 which is 3 then n equal to 3 to 2 3 minus 1 is equal to 7 you will notice that 5 has not been obtained 2 to the 5 minus 1 it's 32 take away 1 is 31 2 to the 7 minus 1 is again prime 127 but if we raise 2 to the 4 2 to the 4 minus 1 is 15 which is not prime 2 to the 6 minus 1 is 63 again which is not prime so it turns out that the exponent n must be a prime for 2 to that power minus 1/2 B Prime so that is a necessary condition the power or the exponent has to be prime but it turns out that it's not sufficient so we have to have the exponent of the power being a prime but it's not good enough to guarantee the 2 to that power minus 1 is prime and an example of that is 2 to the 11 if you look at the middle of the slide not all prime n make 2 to the n minus 1 prime it has to be a prime number but as I say it doesn't guarantee that it gives you a prime because 2 to the 11 2 to the 11 minus 1 is 2047 which you've quickly factorized as 23 times 89 but they are useful because there's a very straightforward computationally straightforward still takes time to check whether enough whether or not a number of this form is prime so people who are hunting for the largest Prime's find so far I've got that one at least that's what it was when I lived up on the internet on Sunday it's 2 to that par there another it's got 13 million digits sorry this is wrong and it doesn't have 30 it has 10 thousand billion digits I ten thousand billion digits and what I meant to write in that last line is that if you were to write it out on a strip of paper okay some of you may have smaller handwriting than me or larger handwriting or whatever but it goes for about a million miles a big number it's not known whether or not there are an infinite number of mercy and primes there's an awful lot you can say in number theory that is not known and they certainly don't give all the primes you may have noticed the method did not give five skipped over five but there is a remote remarkable expression for generating Prime's not only generates primes but it generates all of them and that is this and I always wanted to say this consider the following polynomial in 26 variables a b c d up to zed and that is it taken from a worklet the previous professor of geometry Robin whom delighted to see Robin Wilson delighted to see in the audience and I worked on together not this result sorry a book in which this result appears and so k wz c wz + h + j- g or square and what you do is you let the variables a b c d of 2 xored run through all the non-negative integers give them all non-negative values whatever want you happen to choose and sometimes this expression will evaluate out to be a positive answer and sometimes it'll value we write to be a negative answer the amazing thing is that whenever it's positive its prime and even more amazing every prime can be obtained in this way doesn't really help you very much but it's it's one of those amazing things that can keep you awake so the positive values as it says there are prime numbers and every prime number can be obtained in this way all right now let's look at what happens when we start adding or subtracting primes this gives us some of the most intriguing difficult but easily stated problems in mathematics for example if we add two prime numbers other than two we always obtain an even number that's because every prime number apart from two is odd and if you are two odd numbers together you get an even number but can we get all even numbers in this way that's the Goldbach conjecture can every even number griddling for be written as the sum of two primes and here at the time some examples it is equal to three plus five ten is five plus five it's also 7 plus 320 or 7 plus 13 so 17 plus 3 200 is 7 plus 193 2040 is 119 plus 2021 this has been checked up to an amazing number 4 by 10 to the 18 that's 4 followed by 18 zeroes and it's fine to be correct but nobody has any idea as to whether it's true or not for every even number that's what I meant by difficult but yet easily stated so Goldbach's conjecture is can every even number grism for greater than or equal to 4 be written as the sum of two primes quite remarkably there is a result that's I think the best part best we have so far due to Jing Ron Chen who proved in 1966 that all sufficiently large even numbers that is all even numbers big enough are the sum of a prime and the product of at most two primes so far enough out into the integers there's a point beyond which we can prove that every even number can be written as a prime plus at most the product of two primes and we turn to another result the Goldbach conjecture was concerned with adding Prime's together and we obtained another tantalizing conjecture if we think about subtracting primes now the only Prime's that are 1 apart are 2 & 3 because all the other primes are odd so must be two or more apart but how many Prime's are exactly 2 apart and a twin pratt twin primes are a pair of primes which differ by two and if you list the byte 3 5 first prime 4 5 7 11 13 17 19 29 31 and got up like that 100 709 2027 2029 1 million and 37 1 million and 39 other infinitely many such pairs after all you saw almost a one slide approach proof to show that there were infinitely many Prime's but nobody knows whether or not there are infinitely many prime pairs and up to 10 10 to the power of 16 there's that remarkable number of prime pairs but the belief is that the primes like London bosses can come in pairs but they don't common threes Prime's don't come in threes and a prime triple except for one exception is a collection of three primes of the form and n plus 2 and n plus 4 and the only prime triple is 3 5 & 7 all right and the proof is homework which will not be marked but I've given you a hint whenever n is not 3 then one of the numbers and/or n plus 2 or n plus 4 can be divided by 3 and if one of them can be divided by 3 of course it's not prime now that's the question that used to be asked at at interview all right we've seen that the primes occur forever so as we walk through the integers we're always guaranteed to have a prime ahead of us we've also seen that it's difficult to say where that prime is going to be but we know there's going to be one and indeed it seems likely that there's going to be a twin pair ahead but it's not being proved that twin primes twin pairs occur forever on the other hand we can find a run as long as you like were on our walk where there are no Prime's you have to go quite a distance before you find such a run but you can't find it and that's a neat little argument that makes use of the factorial function and here the factorial function takes a whole number and 2014 factorial is 2014 multiplied by all of the integers less than it so let me do it for an easier when factorial 3 is 3 by 2 by 1 factorial 4 is 4 by 3 by 2 by 1 factorial 5 is 5 by 4 by 3 by 2 by 1 so the thing about this very large number because it's all of these multiplied together is that it can be divided by 2014 it can be divided by 2013 it can be divided by 2012 2011 and you're getting the pattern it can be divided by 4 by 3 by 2 and by 1 so now let me construct the following run of numbers there's this big number 2014 factorial plus 2 that number plus 3 that number plus 4 that number plus 2014 if you look at the number it can be divided by 2 because 2014 factorial can be divided by 2 it can also be divided 2014 factorial plus 3 could be divided by 3 because 3 divides under 3 and 3 divides into 2014 factorial the next one could be divided by 4 because 4 divides into each of the terms the 4 and the size 14 factorial right the whole way down to the last one in the sequence so there we've got a collection of numbers a run of numbers there are 2013 of these numbers I chose that because it's this year's death so 2000 and you sought might seem somewhat random and none of these 2013 numbers is prime so if you give yourself an integer no matter how I beg you can do this take this approach and you'll find a run of numbers of that length where there are no prime numbers it may be way it will be way way way up the integers but there's a run of that length with no primes the number theorist Don Zahra at his inaugural lecture at Bourne University in May 1975 said there are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts the first is that despite their simple definition and role as the building blocks of the natural numbers the prime numbers belong to the most arbitrary and ordinary objects studied by mathematicians they grew like weeds among the natural numbers seeming to obey no other law than that of chance and nobody can predict where the next one will sprite but then he went on to say the second fact is even more astonishing for it states just the opposite that the prime numbers exhibit stunning regularity that there are laws governing their behavior and the days obey another they obey these laws with almost military precision but to talk about the second fact we need to change our point of view and we need to change the question that we're asking instead of looking at individual primes or a formula for primes we can't harm any Prime's there are up to each integer and see if there's any pattern in the results so I need to introduce the prime counting function PI of X now the PI here is traditional but it's got nothing to do with the number pi that arises as the reissue of the circumference of a circle to its diameter I've done some calculations of the first few values pi of 10 is to count the number of primes that there are up to 10 and there are four of them two three five and seven pi of twenty is equal to it as their it Prime's up to twenty two three five seven we have already 1113 19 and 19 and pi of 100 is 25 and this is from the sieve which it took me a while to draw so I wanted to use it again and there are 25 Prime's up 200 so what I want to look at now is to represent that graphically and to draw a graph of PI of X for values of x from 1 up to hundred and you get the stir case type of function and as you go up this staircase or go along this Turkish you go up one every time you come to your prime that's why it's got that jagged impression that's why it's got the jagged feature so it begins at 0 jumps by 1 at every prime number so jumps by 1 or 2 and a 3 and at 5 and 7 and so on so although there are oscillations the function grows quite regularly no I I want to show you the graph of that function not between 0 and 100 but between 0 and 50 thousand that is quite amazing and don't soggy who says that the smoothness with which this curve climbs is one of the most astonishing facts and mathematics in the next bit I want us together joint enterprise to find a formula or postulate a conjecture a formula describing that curve so let's go table may look a little daunting but I'm going to take us through it in a gentle manner down the left-hand side I have the values of X and we're going to be counting the number of primes less than X and what I've done X starting at an then a hundred and then a thousand then 10,000 100,000 a million ten million hundred million a billion and so on so you can see what's happening there we're looking at patterns I'm increasing the value of x by 10 each time now the next column is the one where I did all the work I have counted all the primes up to the value of x and the left-hand one and it was easy to start off with 425 168 up to 10,000 third 1229 and then it got a little bit harder nine thousand five hundred ninety two seventy a thousand four hundred ninety eight so this is counting the number of primes and up to the last one there which is ten billion there are four hundred and fifty five million fifty two thousand five hundred twelve now that took the time to get that done can you imagine just how many hours tedious ours were put into put into doing that there so this is like experimental mathematics that we're doing right so if we you know started say we're looking for what happens for X as X gets bigger and bigger I should tell you rather than what happens tone around the low levels so that up to about a thousand the third column is telling you what the proportion of primes are so up to about a thousand a sixth of the numbers are prime that's what the six is in column three up to ten thousand and it of the numbers of prime a plate up to 100,000 one tenth of the numbers are prime up to the next number which is a million a twelfth a fifteenth up to 10 billion one 22nd now what we see there is something very interesting in the third column as X is getting larger look at the numbers can you see any regularity in them not at the start but once you go from about 10,000 you go from eight point one to ten point four do your subtraction you go from ten point four to twelve point seven do the subtraction have you seen that answer before from twelve point seven to fifteen subtract them from fifteen to seventeen point four not quite the same each time it's going up by two point three about two point three so when x increases by ten the proportion of primes goes up by two point three now that's something that you may be familiar of fit with that's exactly the property of logarithms that if you increase something by a multiplicative factor the logarithm increases by adding something on and I've calculated in the fourth column what the logarithms of X are to the base 10 because they're very easy to do because to the biggest an what a logarithm is it's just what power of 10 is it so the logarithm of 10 is just one it's just one part of 10 the logarithm of 100 which is 10 squares just to the logarithm of 10,000 is just 3 because it's 10 cubed and so on so that's all the fourth column is but it turns out that the logarithm we have to take the base 2 is not tan but it's this number e the base of natural logarithm is just another number just not 10 and then you see that if we take the logarithms to the base e you get the fifth column and the fifth column is just the fourth one multiplied by two point three so that's all it's happening as you go down this column here it's just the column to the left multiplied by two point three one times two point three two times two point three Prime all right you've been very good very patient we have two columns I want you to pick out the third and the fifth the third is the proportion of the primes up to a certain number the fifth is the logarithm of that number and when you look at the two columns would you say they're in pretty good agreement here we have ten point four eleven point five twelve point seven thirteen point eight fifteen sixteen point one seventeen point four eighteen point four 19.7 20.7 22 and 23 and in fact the prime number theorem this is experimental Muhammad X the prime number theorem of just pecked i2 those two columns this is the third one this is the fifth one was conjectured by guys and it says that the ratio of the two numbers the number in the third column to the number in the fifth column goes to one as X goes gets large so the reissue of the logarithm to the proportion of Prime's goes to one and we can rearrange this to get algebraically if you multiply above and below by PI of X and divide above and below by log of X it can be just reexpress this PI of X over X log X because it's the PI of X that we're concerned about and you often see the prime number theorem written this way PI of X then this symbol here just means that the ratio of the two things goes to one as X gets larger and larger and larger and gosh conjectured this when he was 15 he studied Tibbles of primes and during his life he once wrote how he would very often use an idle quarter of an hour the coins through another chiliad that is an interval of a thousand numbers here and there until finally he had listed all the prime numbers up to three million and compared their distribution with the formula that he conjectured conjectured namely that the reissue of the numbers of primes Optio value x divided by x over log X approach one the theorem however was not proved and until 1896 when it was established independently by the French mathematician Hadamard and the Belgian mathematician de Ville a person quite amazing to have conjectured that at the age of fifteen but you see it's an experimental result where he was living at the ratio of the two things and if you you had the pattern then you can see you can see what do hard to go on later on he improved his conjecture somewhat oh no sorry I wanted to what Legendre did first of all because you were probably irritated the way I was when I first saw that and that there's a sort of obvious way to make there be a better match I said to you you know look how close these two columns are fifteen sixteen point one seventeen point four eighteen point four 19.7 kind of like me you were probably dying just to take one of this here in order to get a better agreement and that's what we show hundred it his would taking decimal points the factor was one point zero eight three six six which gives you a better agreement but more importantly was a guy or a refinement of guys to it here's what he originally had this is saying that the proportion of primes PI over X PI of X divided by X looks like one over log X and he later had a better approximation here which is what's called for those if you know the calculus is called the logarithm integral but away but an interpretation of this which turns out to be quite important to to conject your estimates is that this is beginning this can be interpreted roughly is saying that the probability that a certain number is prime is 1 over the log of the number now when i say roughly that's because it doesn't really make any sense because a number is or is not prime but the way that this is coming out is that in the sort of interval concerning a number or surrounding a number the probability that you'll find a prime over there is 1 over the log of the number i'm not that as i say that can be very useful in conjecturing estimates ok so now we have the prime number theorem and i want to finish with introducing the riemann zeta function and its connection with prime numbers then you can start on your work to win a million dollars well first of all i want to begin with a remarkable observation of Euler but first of all remember the fundamental theorem of arithmetic that every whole number could be written as a product of primes and only one way apart from the order in which they're written so we've had that and then I want to look at a particular series the harmonic series so-called because of its relationship to music and overtones and and harmonics and Euler had the this most marvelous observation which you might think looks absolutely horrible when you first see it and then you will think that it's so beautiful and pretty when I talk about it it expresses this harmonic series as the product of some other series the harmonic series as a product of other series I know it looks terrible doesn't it but the time it took to put this onto PowerPoint is in itself a marvel now remember your fundamental theorem what we have got down here is a series which consists of the sum of the squares of the reciprocals of the prime to 1/2 1/2 squared 1/2 cubed 1/2 to the fourth here you've got exactly the same thing for the prime 3 the sum of the reciprocals of the different parts of 3 here you've got exactly the same thing for the prime 5 for the prime 7 and for the prime 11 why are the two of them equal well pick any number in the first series at random I have picked 1 over 42 because that's the one that happens to be in blue 142 what are the prime factors of 42 2 times 3 times 7 so what we do is we take the first term in the first series sorry the second term in the first series are half the second term in the second series 1/3 we take 1 from this next series we take a seventh from the series with a 7 and we take 1 from all of the others and when you multiply them together you get 1 over 42 just to show you that it wasn't a trick I shall do it with another number chosen at random which is 1 over 80 on this time it 80 is 5 times 16 5 times 2 to the power of 4 so how do we get the 1 over 80 right of this product of series well we want a 1 over 2 to the 4 here we take a 1 from the next one we take a fifth from the series for the 5 we take a 1 we take a 1 okay so that we've got an identification here between this series and the series the harmonic series in the top honest as a product of series here sorry the pointer is so close to thee there we are to product of series one for each particular prime alright and that was the one that I showed you the one over Eddie now what you can actually do is to show what the sum is of each of these series and for those of you who have seen it is called geometric progression for those of you who haven't seen it before it is very magical and we want to be able to sum this series for example 1 plus 1/2 plus 1/2 squared plus 1/2 cubed and so on so you write down let s be the sum of it then you multiply both sides by X right and something magical seems to happen the 1 multiplied by X becomes X the 1 over X multiplied by X becomes 1 the 1 over x squared becomes 1 over x the 1 over x cubed becomes 1 over x squared 1 over X to the fourth becomes 1 over X cubed and you're nearly about to say to yourself aha but what about this expression down here that's just the series that I started with so multiplying the original series by X gives you x times s equals x plus itself again and you rearrange that to get s equals x over X minus 1 now the point of showing that it is nice but showing it was really to reinforce the point that the original harmonic series is very intimately related to a certain expression involving primes so if I apply that to the harmonic series I get the prime 2 over 2 minus 1 3 over 3 minus 1 5 over 5 minus 1 7 over 7 minus 1 times 11 over 11 minus 1 the point is the harmonic series that I want you to just hold in your head is related to the primes the details of the relationship or not are not important at all okay I'm a natural fact it this this harmonic series in fact diverges again it's almost infinite it keeps getting as big as you like by taking enough terms and in fact this is another way of showing that there are an infinite number of prime numbers which I won't go into at the moment but I want to keep developing this pattern if I may and when we develop the pattern we can we looked at the sum of the reciprocals of the natural numbers we can look here at the Riemann zeta-function this is the definition now the first stage of the definition of the riemann zeta function for an integer k and what it is is the sum of the reciprocals of the k sparse 1 over 2 to the K 1 over 3 to the K 1 over 4 to the K so the next slide will show you what the zeta function that's the Greek letter Z 2 there evaluated at 1 is that's just the harmonic series 1 plus 1/2 plus 1/3 plus 1/4 plus 1/5 Zeta at 2 is down there it's 1 over 2 squared plus 1 over 3 squared plus 1 over 4 squared plus 1 over 5 squared and in fact Euler actually evaluated this by slightly dubious means to get the answer pi squared over 6 where pi is the ratio of the circumference to the diameter of the circle quite remarkable alright and here again what I want to do is to show you the sium approach can be taken to Zita of 2 namely that it could be written out so the thing here the details don't matter even though they're so nicely presented that we're expressing Zita of 2 here in terms of expression involving the primes that's the crux it's can be rewritten in terms involving the primes ok which brings us to the general Riemann zeta function here I've in fact and done exactly the same thing for zeta evaluated a K where it can be written in terms of expression involving the Prime's so the only thing I want you to take away here is that this Riemann zeta function so far can be connected up with prime numbers and I went through it in reasonable detail with the case of the sum of the reciprocals remember one over 42 and the one over 80 and you can do it with whatever power you happen to want to take beneath the line that's really been the summary of the last two slides all right I turns out what Riemann was able to show in a paper in 1859 he only wrote the one paper number theory 1859 spite ten or eleven pages long and you can get the original in German on the web and you can also get a translation of it on the web he was able to extend this definition of the zeta function not only to the integers but in fact extended to all of the complex plane and the complex plane just consists of numbers you know a plus bi where I can be a real numbers I is the square root of minus one and when you're manipulating with it just every time you see I squared you replace it by one and he was able to extend this definition of the Riemann zeta function to all of the complex plane with the exception of a s equal to one because when it was one that was that harmonic series and if you remember I told you that it without proving it I told you that it diverged so we've now got a function called the riemann zeta function defined on all of the complex plane with the exception of the point one and you know what it looks like at the integers 2 3 4 5 etc because that was the what I did on those previous slides so you know exactly what it looks like at those integers and you've got in your head don't worry about the details it can be written out in terms of prime numbers ok so this function capture something about the prime numbers and Riemann hypothesis was that he obtained an explicit formula an explicit formula for the number of primes that there are up to X in terms of the zeros of this riemann zeta function he obtained an expression for the number of primes up to x that's at pi of x remember we had the prime number theorem in terms of the zeros of the riemann zeta function and for the zeros of the read of C to function there are what are called the trivial ones which lie along the negative integers of - 2 - 4 - 6 - it but the zeroes that he uses in the expansion in the expression of the number of primes up to X all lie and what's called a critical the critical is called the critical strip in the complex plane which is the numbers between 0 & 1 the area the complex numbers whose real part is between 0 & 1 and again if you want to do some calculation of what the zeros of the prime of the Riemann zeta-function are you find that the first few zeros are their symmetric about the horizontal axis 1/2 plus fourteen point one I and a half - fourteen point one I 1/2 plus 21 point O 2 and -21 point O two and a half plus or minus twenty five point two one and a half plus or minus thirty point four one and the pattern that you probably see is that the real part of each of these is a half so all the zeros where the Riemann zeta function is zero I've got real part 1/2 and the Riemann hypothesis is that all non-trivial zeros lie on the line x equals 1/2 and it's been checked and the first 10,000 billions heroes do lie on the central line and it's generally believed that all the non-trivial zeros lie on the line but nobody's been able to prove it after a century and a half and although the Riemann hypothesis may seem very technical and indeed it is it's exceptionally important in mathematics and it's truth would give us much better results about prime numbers and that's not only important for mathematics but it's also important in e-commerce because in modern cryptography public key cryptography which I think will be the topic of one of my talks next year factorizing large numbers enter their constituent primes is crucial in modern security systems such as online shopping and online banking so it's not just of theoretical interest as great commercial importance and the Riemann zeta-function also seems important in quantum physics where the spacing between the zeros of the riemann zeta function looks similar to the spacing between the energy levels in a quantum system and thus marcus the so toy says at the end of his book on the music of the primes the primes are central to the security of the modern electronic world and their resonances with quantum physics may have something to tell us about the nature of the physical world and many people explore the primes in different ways but the mention of music made me want to let you hear this audio file of the music of the primes it was created by the number theorist Jeffrey stop while using the audio package in the application Mathematica and it is contributions of the first hundred zeroes of the Riemann zeta function are that one of the time at one at a time in an interval of 9.2 seconds each note has the same amplitude in frequency as the corresponding term in Riemann's exact formula I haven't shown you the exact formula but each coming from a single zero of the zeta function and then at the end all 100 zeroes play together for ten seconds and if q and that's just an advertisement for my next lecture our average is typical so thank you very much for coming just hit it on the botton but if anybody wants answers any questions do come up afterwards I'm very happy to stay around thank you

Channel: Gresham College

Views: 126,917

Rating: 4.8383312 out of 5

Keywords: Maths, History, Numbers, Primes, Algebra, Euler, Gauss, Mathematics

Id: lzyWL1LTlq4

Length: 60min 20sec (3620 seconds)

Published: Thu Jan 31 2013