Ken Ono - The Riemann Hypothesis (March 14, 2018)

so to celebrate pi day I would like to talk about one of my favorite topics and for experts in the room believe it or not I'm not going to be talking about partitions I'm gonna be talking about a subject that we all learn from a very early age just basic numbers and the distribution of primes so this lecture is called the Riemann hypothesis let me just dispel make sure no rumors get started where there will be no announcement about the proof of the Riemann hypothesis so feel free to leave now Peter wants to get up and go I think we will have something interest interesting that I can offer you at the end that shed some light on how to think about these problems but this lecture is really about what is the Riemann hypothesis and why do we as number theorists care so much about it so if you want to win a million dollars I recommend that you think about the Riemann hypothesis because it can be really hard to win a million dollars so it's I stole this image from the internet it's I presume it's from the television show Who Wants to Be a Millionaire and there are some easy questions that could help you win a million dollars for example which of the following is the largest a peanut the moon and elephant or a kettle all right all right so I actually cut off the top of the face because um yeah that wouldn't be very good my name would be not mud with someone and it can be really hard to win a million dollars because about eighteen years ago a collection of mathematics problems called the Millennial prize problems were proposed and each of them comes with a bounty of a million dollars and the this lecture is about the fourth problem the Riemann hypothesis there are many problems that I think are quite well known and this lecture happens to be about the fourth one is probably most of you know the plonker a conjecture has now been solved but apart from that I think these problems are all worth thinking about and they're all very important for very different reasons so in terms of some history the story of the Riemann hypothesis has lots of interesting stories associated with it so here's one of my favorite mathematicians it's an image of G H Hardy one of the very few images that exists and Hardy had has a very famous quote there's a very famous story about Hardy and in relation to the Riemann hypothesis and and it goes like this Hardy once on a trip to Denmark sent a postcard to his good friend Harold Bohr and on this postcard he wrote have proof of our H postcard too short for proof so there's not very much else that has to be said but if you don't know let me tell you the story what was Hardy thinking Hardy had this he had this long life long battle with God and his idea was that God would not let the boat sink on the return journey and give him the same fame that Firma had achieved with his Last Theorem here's another very very well-known quote this is another one of our most important historical figures in mathematics this is an image of David Hilbert and this is a famous question that he raised if I were to well he hasn't this hasn't happened yet if I were to awaken after after having slept for a thousand years my first question would be has the Riemann hypothesis been proven good we have a long way to go we're not even halfway there but this is this is some of the lore that goes along with the Riemann hypothesis so what is the Riemann hypothesis it was postulated in 1859 here's a one of the few pictures I know of of Riemann and stated verbatim it's that the non-trivial zeros of the Riemann zeta-function should have real part equal to one half and that's it there you go so the talk isn't quite over yet so part of this talk will be to explain why that matters the zeta function is and tell a lot of stories along the way so what does that mean and why does it matter it turns out it matters for many reasons almost everything I'm going to say in this talk is a is a prototype a special case of a much larger theory a much larger theory of what we call l functions and and there are big theories that are being developed you may know about the words Langlands program bertson's Winterton dire conjecture and and other things and it's one thing i have to say in case I forget to say that is that a lot of those cutting edge theories have their origins and some of the things that I'm going to talk about today with respect to the zeta function so riemann posed his famous hypothesis because he wanted to study prime numbers so here's a definition a prime is a natural number greater than 1 no positive divisors other than 1 in itself and here's the theorem it's one of my favorite theorems I teach it in introduction of proofs class at Emory almost on day one it's the fundamental theorem of arithmetic and it's the statement that every positive integer greater than one factors uniquely upto reordering as a product of primes so 6 is 2 times 3 and as we all know that's kind of the end of the story however if you start to delve into other areas of number theory there's a field called algebraic number theory I regret to inform you it's not so simple but for us today we will be content with studying the distribution of primes and the primes are an infinite sequence of numbers that begin with these terms 2 3 5 7 so on and so forth I shouldn't have said that it's an infinite sequence of primes yet because that's what the first question you would ask are there infinitely many Prime's but before I get that I would be remiss if I didn't recite one of my favorite quotes about the prime numbers Prime's are ordinary at this end and at the end of this lecture I'm gonna discuss some joint work one of my co-authors who's actually not very good at being the co-author part my friend Don's a gay has a very famous quote about primes and it goes as follows primes grow like weeds seeming to obey no other law than that of chance nobody can predict where the next one will sprout there's another part of this quote but primes are even more astounding for they exhibit stunning regularity there are laws governing their behavior and they behave these law they obey these laws with almost military precision the two parts of those quotes seem to be in contradiction and but that is not the case and part of what I want to do is explain how these two quotes these two parts of this one quote come together and begin to illustrate what's so beautiful about this function called the Riemann zeta function that riemann defined in the 19th century so there is a very simple algorithm for listing primes up to a given bound so let me just recall it for you to remind you that primes have been studied for both since and since antiquity there's the famous soup of Aristophanes here's an image of Aristophanes and a picture basically illustrates an algorithm for listing the primes up to a given bound here's this picture the numbers that are boxed they are the primes 2 3 5 7 all the way up to 47 and the way that these numbers are obtained is by first listing the numbers 1 through 50 if you have the patience to do that you recognize that one isn't prime after 2 you strike out all the even numbers leaving 2 you notice the next number 3 is prime because you haven't struck it out but then you strike out all the multiples of 3 they're on after then you discover that the next number after 3 that has not yet been struck out as 5 so on and so forth and this method allows you to fairly rapidly list algorithmically the primes up to some bound but it's not so useful it doesn't really reveal much about primes as a process it's an algorithm and if we want to know some of the deepest things of how Prime's are distributed although there's a beautiful and deep rich theory in analytic number theory based on ideas like this it's called sieve firry we need a little bit more so the first clue to this believe it or not actually is in the famous theorem of Euclid that there are indeed infinitely many primes and this is a great theorem I'd like to teach this theorem at Emory in my introduction to proofs class because it's a proof that everyone gets and it's immediately a proof you can criticize because you can begin to ask questions about what the proof does not reveal it's actually very interesting really what was Euclid's proof that there are infinitely many primes this is not my typing I cut and paste this cut and paste it after copying this from Google somewhere but the proof goes like this is famous suppose that there are only finitely many Prime's and let's order them p1 equals 2 p2 equals three all the way up through prpr i am going to declare is the last prime there are exactly R Prime's what you could do then is to find this number capital P to be the product of this complete list of our primes and add 1 and because that we had the fundamental theorem of arithmetic that number cannot be prime so it's going to be divisible by primes but that's not okay because that number that alleged prime that would divide this would have to divide into 1 and that's just not okay there's a lot that you can do with this in the beginning introduction to proofs class you can compute some of the first few numbers of the form the product of the first are Prime's plus 1 and ask students are these numbers always prime when aren't they prime and it's a very big very good set of exercises that you can assemble that that that I enjoy giving all right so if we fast forward a little bit to Euler this is where the story really starts you see Euler and this is a device that appears almost in every lecture here at this conference is a conference going on this week on string theory and moonshine and number theory we find that we can't escape something called the Geo Metro series which you learn about first in calculus if R is a real number with absolute value no greater than 1 then the geometric series in our summing up 1 plus R plus R squared off to infinity can be more conveniently written as 1 divided by 1 minus R this kind of identity is very useful because it allows us to express two kinds of quantities in many different forms where one side is useful to us and the other side is actually calculable so just by making use of this geometric series you can write down some strange infinite series expressions which maybe don't seem so strange but but humor me here in this box we have some perfectly good ordinary rational numbers two three fifteen fourths and thirty five eighths which most of us would agree should be written in those forms but what if you didn't want to write in those forms what if we lived in a universe where the right way to write to was to write it as the sum of reciprocals of powers of two if you were in high school calculus you probably would have objected but it's something that you would have liked maybe certainly thirty five eighths is already almost an objectionable fraction certainly probably don't or right thirty five ace is a product of rational expressions like this the point being these intermediate expressions are all of the form that appear on the right hand side of the geometric series and if you formally write them out as a product you realize by the fundamental theorem of arithmetic all we are doing is summing summing up the reciprocals of positive integers in the first case which are powers of two in the second case which are just products of twos and threes and in this last case there's some reciprocals of those integers that aren't divisible by any prime larger than seven so I think most people would agree thirty-five ace is something you can wrap your mind around this is something you might not want to write down but that's the kind of thing that we want to do here so this is Euler's idea and using the fundamental theorem of arithmetic all I've said is that if we sum up the reciprocals of the positive integers n summation 1 over N to the S formally I can describe that as a product over primes now we know there's infinitely many of them and what is it an infinite product of these rational numbers 1 divided by 1 minus 1 over P to the S when we choose s to be a nice ordinary integer now today is pi day so most of the meth addictions in the room can probably guess what's coming next but now you know what's coming next possession said today is pi day so there's some very very beautiful formulas in the spirit of today that relate these expressions with pi so for example if you let s be 2 so on the left hand side that would be the same as so many of the reciprocals of the squares this expression is a famous expression for pi squared over 6 and it's very well-known that if you replace s by any positive even integer where you're summing up through precipitous of 4th powers or 6 powers so on and so forth that there are similar expressions we're on the right-hand side you get the corresponding even power of pi times some rational number which is very interesting in mathematics in fact that's the beginning of many different kinds of mathematics many with very fancy names so let's revisit Euclid's proof on the infinite infinitude of the number of primes and let's not try to be good at it but let's see if we can get some mileage out of this infinite product just to get a sense of why it's a good idea so here's an elementary theorem so please don't be impressed by it but it's a it's a good step so in this lecture PI of X will denote the number of primes less than or equal to X so PI of n is number of primes up to n and let's just sketch a proof of a theorem let's show that PI of an grows at least as quickly as a natural log of n so what I'm actually going to prove is that PI of n is greater than negative 1 plus a natural log of n but it's a good exercise to go through to illustrate euler's ideas you see the idea is that we want to somehow access the number of primes up to a large number X by making use of Euler's function so what's one trick for doing that so here's an elementary exercise that we could go through so as before let's list the primes in order P one is to the first prime P two is three the second prime P three is P 3 is five that's the third prime let's let them be the primes and order and because the primes aren't as common is just counting 1 2 3 up to infinity it's very obvious that the J's prime is at least as large as J plus 1 right not every number is prime so I'm throwing away a lot of information there but let me just pretend that that's like an idea and you'll see that I've actually thrown away a lot but I I'm gonna get a lot out of that so calculus tells us that the natural log of n is ass definite integral from 1 to N of 1 over X DX now calculus also tells us that we can estimate this integral and we can estimate this integral by making use of the the idea that definite integrals give you areas bounded by curves so if I want to draw a picture of what the natural log of n is this is what that picture would be and your high school calculus teacher will tell you about the theory of Riemann sums how appropriate and what you can see is that you can estimate from above this integral by adding up a bunch of rectangles whose heights are in order the sums of reciprocals of the positive integers so of course from this picture it follows the natural log of n is bounded from above by the sum of the reciprocals up to n and if we make use of Euler's idea where we just declare that pi of n is equal to K the last prime up to n will be the K prime all then all the integers up to n none of them will be divisible by any Prime's larger than the K prime and so Euler's idea tells us that the natural log of n is bounded from above by this strange expression despite the fact that thrown away a lot of information now what we want to do in number theory is we want to take ideas like this and this is not good this is not the best way to do it as we'll see in a moment riemann gives us the best idea but even right here the idea is how do we access the number K from these expressions and this is where it was easy so when I said earlier that the Jade prime is at least as large as J plus 1 I can I can replace the J prime as a proxy in terms of this bound by J plus 1 and just with a little bit of algebra we get then to the natural log of n must be bounded by this product I no longer have to write down piece of anything because I'm throwing away information I'm trying to access that K and if I write that in a slightly smarter way 1 plus 1 over J is J plus 1 over J and if you multiply that out from J equals 1 to K you get a telescoping product and so by this telescoping you find the natural log of n is less than something like this lots of stuff cancels out comes this K okay I'm very happy so I'm not being smart I just wanted to get cleverly this is not my idea it is a very standard trick I want to access K but K is what I declared to be the last prime up to N so that's PI of N and so I've concluded the PI of n is at least negative 1 plus the natural log of n so just by throwing away load of information and using Euler's idea that summing up the reciprocals of integers can be expressed as a product I have a way of relating analytic functions they're just the natural log function to my prime counting function in it so I've gotten something almost for free all right so now let's fast forward a few decades to Gauss Gauss and I don't know if I'm gonna get this story right but as a teenager Gauss invented a lot of great mathematics in fact Shimon and I are gonna write a paper I guess about how he proves that black holes in case 3 / T cross T to form abelian groups using Gauss's ideas stay tuned but Gauss while studying tables of primes made an astonishing discovery so remember PI of X is the number of primes up to X so just to remind you so what so what would pi of 6 be pi of 6 would be 3 because it's three Prime's up to six if you assemble a table of PI of X say for X being the first few powers of 10 you get these numbers and if you compare them with the numbers on the right X divided by the natural log of X get these numbers and you notice that these columns are quite similar he didn't actually describe this in this way but what came to be known as Gauss's conjecture is more or less something like this if we consider the function ly of X this modified logarithmic integral you integrate from 2 to X 1 over log T DT right Prime start at 2 then Gauss's conjecture based on numerix that he did as a teenager is that the prime counting function should be asked some taught ik to lie of X which is in turn asymptotic to x over log X what is the tilde mean what it means is that if I were to take either of these two expressions and divide them into PI of X and plug in larger and larger values for X the numbers I'd get out ten-to-one actually the task of doing that is actually very interesting so if if you're a student and you want to calculate primes and compare this function with the actual number of primes up to X offered by and compare them with what Gallus conjectures you'll begin to see some patterns and you might be retracing some of the observations that many distinguished analytic number theorists have done before you it's actually a very beautiful thing to do I will show you a graph in a moment that illustrates things that offer glimpses of that so this is where Riemann enters the picture he wanted to study Gauss's conjecture he wanted to prove Gauss's conjecture and riemann only wrote one paper and the subject it's probably the most important paper of eight pages in length certainly an analytic number theory ever written and I think in this picture and this paper is actually totally available online if you go to the Wikipedia page I think for the Riemann hypothesis you can click and actually see all eight pages of this paper so I did that and then I cut and pasted images that you can find that are from this paper so in particular here it is 1859 I don't read German but it says something like on the number of primes less than a given large number published in November of 1859 and halfway down that first page you will find this picture this this this function this expression that Euler we that Euler understood the importance of that I've already made use of formally and what Riemann wanted to do is turn this into a function in s where s is actually a complex variable because he understood that if you wanted to make real progress on the prime number theorem Gauss's conjecture probably what you would have to do is understand this function a whole lot better so what's in this 8 page paper he defined the zeta function he determined many of its basic properties and of course he posed the Riemann hypothesis that very strange statement I started this lecture with the statement that this function has the property that it's non-trivial zeros have real part of s equals 1/2 something that doesn't seem so friendly and as we now understand it really is not a very friendly thing but most importantly the point of this paper was that it offered a strategy for proving Gauss's conjecture and the strategy has percolated through modern number theory in many many many different forms so what are some of the facts what are some of the basic properties of this function as you can see there will be four of them and as you'll also see I'm very I'm not very good at PowerPoint so let me just do it anyway all right so zeta function for real part of s greater than 1 we are just going to declare is given by this infinite sum 1 over N to the s is the range of convergence we're very happy with that and so this is the starting point this function has what's called an analytic continuation to the complex plane I've always don't like that because it doesn't mean it is analytic everywhere this function has a pole at s equals 1 but apart from that pole and the in terms of the calculus of complex numbers this function is about as well-behaved as you would want the function to be this analytic continuation is derived in a very awkward way it's derived in terms of what's called the integral representation of this function if I were to show that to you I'm sure you'll probably find that objectionable so let me just say that the conclusion from this integral representation is that the zeta function Zeta of S has a remarkable symmetry property its values at s reflect its values at 1 minus s if you're willing to work with these multiplicative factors and among other things because of the presence of this this gamma is the ordinary complex gamma function and because of that presence turns out there some trivial zeros that this function has and they are the negative even integers ok all right so some of you might know this very famous equation that the sum of the positive integers equals negative 112 I think on YouTube there probably a dozen or so videos explaining this probably some of them explain it exactly in the way I'm about to explain it but if you don't know this it isn't quite true that the sum of the positive integers isn't it isn't positive infinity and turns out to be negative 1/12 but it isn't quite wrong either so I like telling this story because we made a film about Ramanujan and among the many things that he did he made this audacious claim in one of his enigmatic first letters he wrote to gh Hardy and I quote from this letter under my theory one two plus three plus four dot dot dot equals negative 112 and if I tell you this you will at once point out to me the lunatic asylum I like this quote because who talks like this today and it is crazy at first glance but of course two Hardy this would have not been crazy at all so in the context of everything I've just described how do we view this well here's a proof and I've showed this to you earlier when we summed up the reciprocals of the squares 1 plus 1/4 plus 1/9 dot dot that was the number of Zeta of two the two represents the squares and that's well known to be PI squared over 6 if we just pretend and declare that Zeta of minus 1 should be the sum of the positive integers in order which is where we really shouldn't be declaring that but let's pretend if we allowed Zeta of s to be summation 1 over N to the s if s is negative 1 then formally this is what you would get and of course what Riemann noticed what I put REME on there ok any event what Riemann noticed in his functional equation is that Zeta of minus 1 and Zeta of 2 because 1 minus a negative 1 that's what comes up in the functional equation would be 2 are related in this way but aha Zeta of 2 is PI squared over 6 I've got a PI squared here in the denominator and you can carry out the calculations with sign and discover that lo and behold you get out a negative 1/12 so of course from Hardy's perspective that would have been brilliant because very little else and those first few letters that Ramanujan wrote to Hardy offered evidence that Ramanujan knew anything about proper analysis ok all right so what is it that Riemann had in mind in his famous eight-page paper I don't know that it that he could have graphed this but he understood the importance of the values of this function in what's called the critical strip and more importantly he understood the value of locating the non-trivial zeros and I'll get to that in a moment but if you were to graph the zeta function now that as it as we have now defined it where the real part is a half that's fixed and only only graph it in t aspect so the imaginary part is walking up from the origin up to about imaginary part 50 you'll get this beautiful spiraling curve what does it mean for the zeta function to be zero well of course it means it's going through zero and there are many beautiful pictures that you can steal from the internet because I don't have the skills to produce them myself that illustrates the rather remarkable properties of this function and this is just one particular aspect that is of interest to us so what I'd like you to note is data of 1/2 is something like negative one point four six zero three five four so I think that's believable based on this graph and graphing up to T equals 50 you encounter the first few zeros as you see and now I think it's appropriate to revisit what Riemann's conjecture is is that all of the non-trivial zeros apart from the negative evens will be the zeros that are encountered along this path if you're spoiling forever all the zeros would have to be encountered in this way some of you might recognize this picture I think this is uh this is Dennis hedge halls writing so there's a famous image of the location of the zero so if you were to consider the zeta function on the complex plane between zero and one for the real parts the first few zeros are here they coincided with what I showed you in in in that red graph and there are no other zeros certainly up to that height and that was and that's been known for quite some time so riemann made this proposition and he said it would be really desirable to have a rigorous proof of this proposition by the way he wrote in German so every time I've quoted Ramon I'm actually quoting a translation which I assume is correct but this is the understatement of the lecture it would be desirable to have a rigorous proof of this proposition so what I want to explain over the next few minutes is why it would be really desirable to have a rigorous proof and not have it be related to winning a million dollars okay well so let's let's just take a small step aside for a moment and let me show you some graphs of some auxiliary functions you encounter many famous names in analytic number three by just studying this question and work of chebychev and vaughn mangled on sort of illustrates how one should think about the distribution of prime so my guess as many of you have will not have seen this picture I'm about the show but the moment you see it you'll realize why why what what later became known as the prime number theorem Gauss's conjecture kind of had to be right so the prime number theorem is equivalent to the statement that the limit as X goes to infinity of psy of X over X is 1 where psy of X is this very strange function you sum up log P for all prime powers up to X and it's the idea that every time you encounter a prime or a prime power power you you you you indicate that not with the characteristic function as we've done in in PI of X but you indicate that count with its logarithm so if you've never seen this before graph it so every time you get a prime you get this very very unsmooth function it looks something like this so every step up occurs at a prime but what you do notice is that this graph looks like it's doing its very very best to hover around the line or y equals x so I ignored it ignore that which is exactly what the content of this theorem is that size of X and a line y equals x want to do a remarkable job of mirroring each other so if that's what you want to show you want to say then why do the non-trivial zeros matter and this is this is this is what's this is one way that I would explain why it matters it turns out that if you want to show that sigh of x is asymptotic to X you want a formula for sy of X where the main term is X oops I've forgotten this is not a pointer when the main term is X and it turns out that it's a famous theorem in analytic number theory that sigh of X is X plus some innocuous factors but then this infinite sum glaring in red where you sum over the non-trivial zeros row of the zeta function X to the roll over roll you see if that red function wasn't there the statement that sigh of X is asymptotically X is literally trivial because you know you can divide X into sy of X then you get sy of x over actually goes one plus terms which rapidly decay to zero as X goes to infinity but there's no working around the existence of this bold red expression and that's why we really need to know about the location of the zeros oh yeah that lowercase X is the same as the big case uppercase X yeah yeah every X there's only one meaning for X in this lecture should be thought of as a large real number tending to infinity remember when you assemble a lecture by cutting and pasting from the internet and talks right these things things can happen well finally finally the prime number theorem was established in the late 19th century by how to Mardel eval a possum got they prove Gauss's conjecture it is true that PI of X behaves like x over log X and some Tata Klee and there's a little bit heavy but let me just say the proof boils down to the fact that they were able to show that the non-trivial zeros um of zeta our never have real part equal to one all right and that's quite a bit of work I've never seen that before this is this is a prototype for much of what is presently done in multiplicative analytic number theory so why does Rh matter well you can prove the prime number theorem but really not be in a position to optimally use its conclusion so for example if the Riemann hypothesis is true people have worked this out you can actually bound the difference between bound the difference between PI of X and this logarithmic integral which is used to estimate it in fact it's quite good in these there's a famous paper by Schoenfeld among other things one of the conclusions in that paper is that PI of X and ly of X never differ by more than something that's roughly size square root of x once X is at least 2657 and compared to x over log X which is the main term this expression on the right is really really small so this idea of getting a an error term of roughly size square root of x that's kind of like the magic barrier that one is trying to approach in the subject so if you really want to know how accurate the prime number theorem is you want to know the Riemann hypothesis but it turns out that the Riemann hypothesis is one of many Riemann hypothesis there's something called the generalized Riemann hypothesis their Riemann hypotheses for varieties over finite fields there's all many different kinds of them and in the spirit of this lecture let me just restrict to the case where I'm speaking about non-trivial zeros of L functions which are morally relatives of the zeta function and even in those cases it's safe to say and absolutely fair to say that basically every deep question on Prime's we would be informed if someone proved the Riemann hypothesis it's not true that every deep problem about primes would be solved morley pretend that that was true there are statements in in arithmetic geometry related to the distribution of ranks of elliptic curves orders of class groups that are informed by L functions that are very much like the zeta function and there would be huge advances if if someone were to make deep progress on our H if you're interested in quadratic forms module bhargava together with kind of forgotten his name now so I feel really bad about this hope he never watches the video Bhargava following up on a very famous work of Conway HD burger John a very good Jonathan Hankey if you're watching this yeah I'm thinking of you so there's a there there's some very beautiful theorems by Bhargava and hanky what are what are called Universal quadratic forms which gives simple tests if you started with a set of positive integers and you wanted to know whether a quadratic form represents them all it's a finite test they prove that these tests exist unfortunately if you want to actually effectively show how to construct all of these tests and prove that they always work you in many cases still need to know about the Riemann hypothesis surprisingly if we work in algebra we don't even really know about the maximal orders of elements and permutation groups the symmetric group SN as n goes to infinity we are not very good at estimating the maximal orders and that is a problem that is also related to the Riemann hypothesis if you're interested in prime ality testing the running time of algorithms depend on the Riemann hypothesis and I think in in many settings people just use the Rahman apophysis hypothesis and declare that is true in it and it's happy with that but if you really want to be sure that application is definitely there so to summarize these bullet points it's really true that thousands of results are proved assuming the truth of our H and G RH and so any progress on that would kind of like in the most amazing Rube Goldberg experiment result in like thousands of conjectures or several hundred for sure I'd like to give you one I'm giving a lecture here and I like talking about Ramanujan so I wrote a paper many many many years ago with Kenan sounder Rajan who's a very well-known number theorist at Stanford on this quadratic form that Ramanujan wrote down so here is here is his famous quote from 1915 I believe Ramanujan said that the even integers which are not of the form x squared plus y squared plus 10 Z squared what does that mean these are the numbers you can't cannot get by substituting in for x y&z ordinary integers the even numbers which are not of this form are the numbers which are of the form 4 to the lambda times 16 mu plus 6 never mind that part that's easy we know how to work that out the most important part of this quote is what he ends up with he says while the odd numbers that are not of that form it's a set 3 7 all the way up through 391 in his paper do not seem to obey any simple law so just emphasize let's take the number 7 on this list you cannot obtain 7 by substituting in for x y and z an integer in x squared plus y squared plus 10 Z squared in the all of these numbers you can check cannot be obtained a very famous theorem in the early 1980s by Bill Duke and I guess together with Schultz pilots pillow ok said that there that with respect to this list there are most finitely many numbers that complete this list sound and I were very interested in trying to complete this list we were unable to do it but with a little bit of help and this is a little bit more technical than what I would have written here but roughly speaking assuming the Riemann hypothesis for a very specific family of L functions then it turns out that the only positive odd integers you cannot get by substituting in for x y&z are on this list and this list is complete so that's kind of an amazing statement if you don't even have to check that there are any others if someone had to offer a test it the test is here so when Ramanujan said that this quadratic form does not seem to obey any simple law he was absolutely right because what is that law it's basically this this umbrella of the grh and that's certainly not a simple law we wish that was a law but there's a lot of laws let me go on no politics yeah we wish it were a law and we wish it was a law that could not be defied and we wish we could prove it all right so how about some evidence for the Riemann hypothesis well the lowest 100 billion and I think this number changes because people are computing more but I think this is still fairly close to accurate the lowest 100 billion non-trivial zeros so this is ordered by imaginary-part satisfy the Riemann hypothesis so could the first hundred billion zeroes be wrong no no there's a beautiful series of papers and I'm gonna so this is a theorem by Solberg Levinson connery and this this this area of research lives on I think I think Matt Young told me there's a paper on the way where this number will be improved but for now this is definitely what I can tell you that at least 41% of the infinitely many non-trivial zeros for the zeta effect for the riemann zeta function satisfy the riemann hypothesis i think matt said to expect an improvement in the second or third decimal place so this is very hard work very hard work there's a lot of work that's been done on determining zero free regions for the zeta function well the zero free regions of the stuff here that isn't white so if the Z and so if we want all the zeroes to be on this dashed line that's very disappointing to see that the slope of this curve asymptotically tends to the vertical boundaries of critical strip now that doesn't mean that people in analytic number theory have not developed tools for getting around these sorts of difficulties for some kinds of theorems if you ever hear about what are called zero density estimates and the like there are tools that that give approximations to some of the implications of our age what about prospects for a proof for the Riemann hypothesis this all appeared very quickly they're supposed to appear one at a time but I can deal with it if you go back decades much in the early 20th century it turns out that mertens was able to show that the Riemann hypothesis is equivalent to this sort of bound for the famous Mer be as functional Irby is function is a function that it's a very delightful function it's the weight to detect the number one from all other numbers so mu of n is 0 if n is not square-free otherwise it's plus or minus 1 depending on the parity of the number of prime factors and if you compute these sums for large numbers X you never get anything big and despite the fact that the möbius function is very easy to define it's very difficult to show that this sum U of n up to X is bounded basically by the square root of x Big O of X to the 1/2 means just thing doesn't get bigger than roughly the square root of X the reason that that comes in naturally with the zeta function is if you were to reciprocate the product formula that Riemann wrote down in Euler wrote down you would basically be getting a generating function for Mobius and then there's some steps along the way that allow you to make this connection there's a program due to polio which I'll talk about briefly in a moment there are lots of ideas and functional analysis the buzzword certainly around the East Coast trace formulas and and and many other works in particular Peter Sohn knack is here so Peter I have to move on my time is running but there's a very beautiful volume that Peter wrote with cats on random matrices which have implications throughout all sorts of distributional counting type problems not only with respect to primes but in arithmetic geometry and the like alright I would have briefly touch on the random matrices because what I'm going to end with in this lecture is a statement of some theorems that I can offer you and to properly understand them I think I need to tell you the story about Freeman Dyson and Hugh Montgomery and Andrew oh let's go so in the early 1970s I don't remember the year but it's very well-documented Hugh Montgomery visited the Institute for Advanced Study and at the time he had been he had been studying the joint distribution of zeros of the zeta function he was very excited by some of the numerix that he had discovered and in a in a conversation with Freeman Dyson I guess in the tea room at the Institute dyson recognized oh yeah yeah yeah this looks like some calculations that go back to Vigna and others when they were studying the distribution of atoms with very heavy nuclei and that's the very beginning of the subject called the theory of random matrix models for distribution of zeroes and rule it's Co computed many zeros for the zeta function to test their conjectures and in very vague language what they conjectured is much more precise than this something called the pair correlation conjecture but for today let me just say what what they conjectured is that the non-trivial zeros of the zeta function suitably normalized seem to be distributed like the eigenvalues of random hermitian matrices ok all right so what I want to conclude with is some some recent work along those lines and I want to offer a little bit of evidence I think new evidence for the Riemann hypothesis and it's related to the work of yin sand in Puglia it's actually very interesting really Jennsen when he died in 1925 left behind notes like someone else that I study a lot and these notes were passed on to george pólya who was a very famous analyst at stanford for many years and i'm gonna refer to the the papers that came out of this as the y ensign polio program so there's a little bit technical but i'll offer the real details on friday but it goes something like this Jensen's idea was to study the Riemann hypothesis and a whole family of related problems by associating to the derivatives of the zeta function the central ones a doubly infinite family of polynomials with real coefficients more generally for any arithmetic function that say a a of n is a sequence of real numbers for every pair of integers D and n you can construct a polynomial of degree D whose coefficients are the binomial coefficients D choose J times well the terms in the sequence so what is a of n plus J a of n plus J is the J term after the nth term so when I talked about shift n imagine having an infinite sequence of real numbers grouping together like D plus 1 numbers and just inserting those numbers in in order as the coefficients weighted by binomial coefficients why did he do this because he noticed that at least in the case of the zeta function when you insert for the sequence the central derivatives of what's called the Riemann's i function he noticed that these polynomials should be what are called hyperbolic that's very fancy language for just saying that these polynomials have roots which are just ordinary real numbers by the way that's not ordinary a typical polynomial degree 50 doesn't ordinarily have 50 real roots so these are hyperbolic polynomials are not ordinary in a study of hyperbolic polynomials and related things it's actually a very it's a ubiquitous idea what did they prove they proved forget all this other stuff but they proved that the Riemann hypothesis is equivalent to the hyperbola s'ti again all the roots of the polynomials I'm showing you must be real the Riemann hypothesis is equivalent to the hyperbola s'ti of this doubly infinite family of polynomials where the sequence in question are the taylor coefficients of a suitable renormalization of the Riemann zeta-function so gamma Vann literally think of as being the nth derivative at s equals 1/2 for the riemann sigh function which you might as well think is the zeta function well it's actually very hard to show that a polynomial degree a thousand has all real roots so it's certainly going to be very hard to show that all of the polynomials in some doubly infinite family when the degrees go to infinity must be real and so what was known very little was known so in a PhD thesis chaas proved the hyperbola s'ti for when n is 0 for degrees up to two times 10 to the 17th that's a finite set in the 1980s sorta snore folk and volga Fargo and and Dimitrov and Lukas handled the quadratic in the cubic case with quite a bit of work and let me say why it was hard work if you're going to prove that these polynomials are all hyperbolic it's assumes that you already know how to compute all of the derivatives of the zeta function and that is a very big assumption in fact maybe some of the experts can correct me but I think the first proof of the positivity of the derivatives of even ordered s equals 1/2 wasn't even published in literature until like November of last year maybe that's not right but something like that is close to right if you actually compute these derivatives and I'll show that on Friday a lot of these derivatives are like barely positive it could be like zero point thirty zeroes and then like up one two three these this is an insidious function to differentiate and four degrees at least for that nothing was known even computing numerically this was difficult so for these reasons I think most people would think that this is not a profitable line of research which I tend to agree with but it doesn't mean that we don't do it so what is our work this is joint work with my former students Michael Griffin and Larry Rowland together with Don Zog EI and our theorem is this for every degree not just one two and three every degree D could be ten to the tenth any D we have defined a normalization a renormalization for those polynomials called J hat and we take the limit as n goes to infinity of these which is like taking higher and higher derivatives of the zeta function those polynomials actually converge the polynomials that are known and are very much part of this random matrix theory they converge to what are called the hermite polynomials they converge coefficient by coefficient to the to the degree D her meet polynomials and of course the hermite polynomials they are orthogonal polynomials with respect to an exponential measure which by the theory of orthogonal polynomials means that they have many special properties one of which is that they are all hyperbolic and in particular not all they only hyperbolic I know exactly where all the zeros are so the main catch here is that for each degree D because of this limit all but finitely many possible of the derivatives of the zeta function will have a degree D ensign polynomial which is hyperbolic so I offer you this and if we're allowed to cheat and I take boxes and take the widths of the dimensions of boxes to infinity in a suitably but obnoxious way I could say that we've proven 100% of these polynomials or hyperbolic but that would be like politics today there'd be it wouldn't be quite fake there'd be an element of truth to it but what I want to stress is that we've actually located the zero so what is this like saying so it's like saying this it really should be quotes around the word theorem but this is what I want certainly experts to be thinking what this theorem says you see if we prove in this limit and we're able to make it effective by the way how does this how does the hyperbole isset II follow these Fermi polynomials have all real roots and they are all distinct so if you have a sequence of polynomials that converges to such a polynomial with distinct roots at some point it cannot be true that the zeros always have complex conjugate pairs because they do want to double up in the limit and produce a double zero it's just it's just that so what the magic is is it's J hat so on Friday I'll explain how you actually compute these derivatives we do it to arbitrary precision you don't have the right to claim a theorem like this unless you really know to arbitrary precision these derivatives turns out it's not difficult to do we have that and then I'll explain how we get this limits on Friday but for experts I want to stress this this is somehow related to the the conjectures of Tyson Montgomery and follow up by the calculations of a let's go is that I'd like you to think of this theorem is saying that the gue random matrix model prediction is true for the Riemann zeta-function and derivative aspect so let me because Peter is here explain what I really mean by that so the way you think about these gents and polynomials of degree D and shift n is that the degree D shift n hence and polynomials want to emulate the lowest lying D zeroes of the nth derivative the zeta function if the Riemann hypothesis is true then the zeta function lives in what's called the polio Legare class which is closed under differentiation so also gue should hold for all of them and so there is then a gue prediction for the derivative aspect and we show that the hermite polynomials which satisfy the gue distribution which is Vigna semicircular law is free great anyway this is what I want to end with I think this is very good evidence for this body of work related to the Riemann hypothesis we have some evidence supporting now as a theorem a random matrix model and I think for people who study our work they'll be interested in how we actually compute these derivatives and before anyone does that I'd like to say my PhD student Ian Waggoner is going to be doing that to a large class of L functions modular L functions to your cell phone basically L functions with euler product that are that are real that are analytically continued by means of integral representation so these theorems will be true there - great thank you very much [Applause]

Channel: Simons Foundation

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Id: HAx_pKUUqug

Length: 58min 32sec (3512 seconds)

Published: Wed Feb 06 2019