IAS Distinguished Lecture: Prof Shou-Wu Zhang (2 Feb 2018)

so today we are very happy to have professor channel from Princeton and is to give us a talk so professor Jeong graduate from Columbia and visit me be a faculty at the Princeton Columbia several places and eventually now his professor at the Princeton he's visiting us this winter for two almost two months he he's he's a personality arm to be geometry and a number theorist so he has done other work in that area for example in our semantic geometry like Mary Morrison positivity and Pocono of stability conjecture and among others Zakia gross formula etc so here we will talk about L functions he's passed on the future so it will be accessible to almost everybody in at least in the very beginning and also here by the way he's a member of American Academy of Arts and Science okay so welcome it's nice come back after maybe three or four almost four year to see the older friends here and the last time I remember I gave a talk about a congressman Bob llem that's a variation of problem I talked about a progress today I picked up that problem because the problem self is a very elementary is a very very explain but other hand is there that I could normally talk anything over the tools to prove their the conjecture so today I will talk one other - I probably view as the one most important here in a number theory and it's called error functions and how many people heard the error function before just one okay three three yeah so so there is a some people call error function some people call air series people hurt air series and mode because errors it means the Black Series series right so when you buy a car there's air series yeah so I had an exact same problem so we're not going to college at that time number theory was very popular so we I they're the proof of the proof of the 1 plus 2 theorem of which in range hand so I went to college so I window book store I bought a book about a proof overcoat about conjecture then I a few pages they come with the error function and I had no idea about this stuff it's not really nature subject so I came up a subject so I just threw the book away I study algebra so as you were much neat clean no no functional calculus but when I go to the College let's in change everybody use the error functions there's nothing else I can use so I study as well what's help this stuff the secure what people just one studies agribusiness and whether to come from where they will go so the day this is my lecture my discovery my research and the subject why people want to study the same where when the goal okay so so we go back to know this it doesn't work it's freezed oh okay so I go back to the problem you're probably a matter when you in a little super elementary school so the question will be I kept a positive integer K can you find a formula what is the case of power plot T to the kaiser power up to the 100 the case of power right so this is a question where K equals 0 that's easy 1 plus 1 plus 1 100 times yes of 100 that's it is the basically definition then you come to slightly more complicated one plus two plus three of the 100 so the kind of lazy teacher gives you the problem right well for some other kids you find an answer 50/50 buy you some way to do that not only we have a formula we also have the story to tell how girls discover the formula when he was nine years old right so but it but if the teachers more challenge she's here to give you this problem 1 square plus 2 square over 100 square Caillou found some trick to do this problem right now it's not easy to find it well what about super power you keep it going your fun is a much bigger number to do that I don't think I can easily found attracted to this thing this way before you know some substantial mathematics yes there has two tricks for all the powers so well let's say what the tricks meant that we needed to find our formula for arbitrary K arbitrary and try to see if there the formula for these finest sons once the case of power choosing as a power surge could pop up to the end the COS power believe or not even there's a problem you need calculus right ok so that's that really was a history doing K equals 0 studio definition you don't read that much k equals 1 the formula actually precisely given by path occurring in a 550 BC that's or do you know that k equal to 1 square plus 2 square by n square this is a formula we know is attributed to Hakeem it that 250 BC that's 300 is the letter but the first formula where equals 3 that took much longer time your mother is a most is more than 700 years is given by the quit India mathematician baba hotter he figured caco3 and a cake was one are very related you've completed sound is square you get is 1 right that's a really amazing formula well how can I do for arbitrary case I have one K equals 0 1 2 3 4 arbitrary K well I'm Jerry the general formula already over tended by Fermo pascal Bernoulli there's quite a mathematician believe me they thought about this question one way the other way and in fact this they already all of them discover their the general general formula to compute as finest sums up to n and the formula is a pretty simple just one polynomial you just take one more cut power there there's a coefficient you to calculate a k1 k2 a KK this are some rational numbers so this wasn't known but is there this these numbers are very complicated so the question is how can you find a simple formula for that ok this is my experience so when I was in middle school somehow you know you've written some books so one book I read it is a hollow against a book I bought a mathematic induction sedusa book maybe many of people in my generation read this book right in that book this is a wonderful person he he rested that he wrote a formula there that he say prove it right right once you know the formula prove Allah is not a hard problem just use them as Medine that you write ok so this is the whatever this is a book we know in there when we're now with the young so then player probably the middle school right somewhere and they're in the formula I want to say today is in a book or art of conjecturing so for the people study statistics and the probability this is a most important book because this is the first book systematically in reduced probability coming at Horrocks so it is given by jacoby Bernoulli so in the Bernoulli family this is the number one first guy and he wrote a book called art of conjecture in so it's just the art of getting right because you have a probability you can't guess what's going on the conjecture is not in the modern conjecture a means the case right I want a case what I have a next step because I compute a probability right and if he introduced this a Bernoulli number Z is the Z minus 1 the reason is why you only reduce the number because this is nobody I mean I put it easy in the numerator denominator how do you do this in so you just you if you give Z minus 1 you take a tell expansion then you start with Z you can say opposite down Z you start with the one you can use geometric series to take its banner there but anyway you put on them back or the P of J right that's a very power mathematician once we don't know anything to put a number there give a let give a name then there's a fan of some problem has a very simple formula it just arrived on it's very similar binomial formula but it's just instead of a piece out of jail you put a be of lower J if you if you if we make sure you don't matter mister if some season the lettuce we don't forget the J's objeto blow it out here then you get this formula right if you forget the chase down stair up stair you can use the formula so if you say I very difficult for you to remember formula I just put this into there that's what he can okay so so this order Bernoulli Euler Bernoulli I mean there this historic is a father at the Tucson he he wanted one of some study medicine others a study religious but his son that happened to to change his his career in mathematics so this is older per newly this is the famous book he wrote that's a question you're doing if you if you think that is very hard to remember the formula because it involved the pile or binomial coefficient another way to do that they use a young face and yeah a Pascal triangle so the young stranger is about there there they're the powers of sum of two numbers then for this formula I only take a difference so you get a difference of these two seen it get different to think at a number neck you want to this instant here you get a many numbers it's a time first one one number the two number three number you keep a boolean so the relation between there's a sum of powers and Pascal triangle Yahoo s and Y is given by the following calculation you put these coefficient in the triangle metrics you compute a universal metrics you wake Addison that's probably the most easy way to get out the psalm of final sounds okay so at this stage you say well the stories are finished the problem is solved right I already know the general formula for the sums of powers of a number and the star not finished yet because you can ask much more okay that before I do that let's just start over the proof right but this is a math lecture I can't give a lecture without Kiva proof so I haven't give up proof okay so the proof in a comunitario way is always like that you put it in the original function actually that don't stay we have a have an expert come here Tori's you have a generating function as Kelvin right you put inside this is a sound this is some nothing magic we're gonna do the old trick and to do is change the sounds that's always a we do when the changes some then you use the the facts of the tell expands exponential function you will get this step right okay this a topic is the easy topic expanded this stuff and the partner of course were no idea but the Bernoulli just say we're a pattern you don't know anything just given number this is a number of put here you put seen Donnie okay it takes the expansion that's what every proof so you put the formula it's easy okay so the proof is not hard it's neither very little knowledge about calculus right some some say expansion of exponential function right that's already remarkable right to prove there the final some of that in knowledge about explanation function whatever negative power we never asked the kisses to prove compute a negative power that's an ST right you think about you even you want to compute one plus one half plus a while 100 100 how do you compute this thing you have to take a common denominator that's that's that's that's not a bad one to the 100 M suing numbers I can do to something like that 100 numbers how big the Janome they are gonna do it's a very complicated so you suppress the ways now this time we'll really need a numerical mathematics right put in the computer you can't compute the approximation so you go 1 plus 1/2 plus 100 you get 5.18 you computer more one policies for you computer more 1.18 so that's the thing you do but if you study calculus and somehow you can do more if you have a nice sequence of a decreasing function when you have a final sum you always a kind of approximated by integration that's it with something we learned the first principle in the calculus right well the question I ask you is not about approximation about exact formula how can I pass the ball in fact if your function I get a final sum you want to find the exact formula okay so so what I think is that a formula is that if this is the power is a positive power I get a polynomial right I just proved it's a polynomial negative power at only spider polynomial but K it appears some function some smooth foundation if I don't have any requirement is a pretty easy right I keep a bunch of number I just use the smoother line connect them I get a function but if I assume this wire to be analytic or promote condition that is not necessarily easy so the answer actually is a yes you always know to get this some can get a nice function of X so this is solved by Allah well here's the 26 years old so so what I all approve action much marginal for any nice function FX the fun Assam is always a special value or some other nice functions so in other word when we do calculus we make a difference of error a lot of time it doesn't need a error he actually has an exact formula so let's just show you how I solved the problem I am a first extend to do well if you have some integrity from there you get a difference equation right wait I have a difference equation so only thing you do that I give us can you find air s right so that's the question you're gonna do so I solve difference equation I realize expert in calculus what I'm trying to say is that the calculus is a to write for therefore the new trend developed by Newton it's a two for the signs for a line is the two for art I just show you how all are doing first thing I wanna do is a technical expansion for s X minus a 1 but his italics arranging is different than the usual one we do we usually resistant X ranges negative Y is the number this one the variable all right so it is it says exercise is a fixed next next one should be variable so we take this Cal expansion so s X minus s prime of X produced and we put everything down you write is done the e to the negative D s of X oh my god so this he write a differential operator in an exponential way well neither sex was invented by Allah so he must know how to use this end much more prevalent than other people he put this in there this is the differential operator then the difference equation will become this equation 1 minus even fu t sx here so far nothing surprising well where i just rewrite this function and don't get anything this was all right doing all i say well you want to solve this equation you need a universities operate right we don't harm with your operator because they see one of - yield they're saying you started with the Dean I'm gonna say well what are gonna do active ah daddy I'll put a tip back here I didn't do anything but this guy now is a formerly invertible because now with one so I put this in back when I put this in back I could have blew them but back again so this was not doing well now it is become a differential equation to the wonder of it here you call this in right hand side except is a formula so difference equation become a differential equation not by approximation is a body formula so he then the integral Paul sighs this circuit SX this side I went to when you when you can read Abedini the first term is Ephesus so you really the integrate integration is the inverse of the laplacian seek at this equation this is looks familiar because this guy up frozen this guy wasn't known that's the that's the teacher told us in calculus the one but this term we never heard of that right but Allah knows actually that sounds like that happen okay so if we take F X equal x ik a this guy the final sound because a polynomial right furnace ax you can not recover the Bernoulli poly the formula didn't agree he's really computer everything so but this is form isn't much much or not it doesn't mean it doesn't really need a polynomial you can say less than 0 okay the case under zero then the Euler formula after constant compute everything so this is a s- 1/n is a 1 plus 1/2 plus 1/3 Pattison you keep going then you can log-in that integration we get this constant gamma you need a predetermined because you have indian i have an integration never tell you integrate danielson right because you need to have if you have any integral you have you know some singular one in the to determine the constant anyway there's a continuity time in there but otherwise completely explicit so the sound of harmonic functions harmonious sound has exclusive formula is not a pro summation if for s equals Arvika the bigger than Y is very nice so there you have the first term which is infinite sum you get all these a formula so this is this is oil as a sin so oiler really push their the easy cam a some positive power to sum of negative power and he gave exact formula only saying he needed to do next step is a compute is this constant right so well now question became how do you very infinite a son right so now they can become even at some problem right so in 1735 oil I make the first major contribute is a problem you solve is a problem okay because the tune which was a quarter-past problem realized quite a mathematician but he only probability on what he has even aged he prefers him 26 he producing about 28 but this is a at the time he made himself a famous mathematician not the first one but this is a famous problem the pastor prabhasa is a it's a place in Switzerland right where the Bernoulli family Gras paella was a student of the second a Bernoulli Johann Bernoulli so the first plane only check would be Bernoulli well Johann Bernoulli is brother older brother his older brother told him mathematics but at the end is a on oprah's much a stronger so they they enough fight into each other all the time son familiar the family right the younger brother has a son her daniel browser daniel browser is a good friend of Allah but Allah was very mad and pass help if the person never gave him a position so only we want Daniel lab that he came back so he this why I could stay in there in the same Pittsburgh for whole lifetime but anyway this old is a problem immediately make himself very famous she found that a sum of their squares the price of way of six this is not easy to guess and I get some single paisa quid there right and the plus 60 so this is he found is the problem now I show he how can Allah prove the theorem I mean if I say that the Euler proved the first contribution his brief enough can use convert the differential operate that's the basically the truth is doing the second one he's also brave enough was a prayer Vietnam is that he transferred the Stags he figures I exclude 0 X was 0 plus man supply plasma to PI that's we know we're high school middle school this is not a typical fact but if I access the pauline whole polynomial you have this fan now we're gonna do next step you rather plan on with the product of the roots well not rotate well science should be like to drag this but sine of X are also tell expansion now you have a polynomial you have a rooted composition you have a polynomial decomposition you can compare roots on the coefficient that's what I do in a computer the curvy axis sort of power if we do this away what do you get you start with X is sum all these roots this is this side negative side with this side that's that's a proof done right and what do you do what's the early not doing that he's brave enough he sing a sign eggs like a polynomial of course then people challenging him sign X this is not a polynomial in the next few years here can spend a lot of time to try to prove this formula it's not easy to prove for example I mean they're the rules does not eat a meal from uniquely for example you put eggs manage of on either side but you could have some trouble anyway so the illah after many years a lot really proven formula he also found other method approve this so anyway but this is really the first proof it's amazing just he's just briefly enough to think about that sign X is a polynomial so if science a polynomial is probably correct the argument also plotted symmetric function this same because once it get its value we know that there you have an elementary symmetric function over roots related to their to the coefficients right you will get some more formula so that's exactly what I've caught you can't do much a much smarter way you can use a Newton's method to get the powers of roots right it's just too small operation you take their check out there they're there tyrella T of this and then you compute a crayfish and tell expansion you get this formula so for example you get one plus this is false this is the sum of of the next of the fourth power give you PI for tip about 96 power 945 the question is that I mean it's only the only I can do even power it doesn't know how to do even hard are apart because this method cannot be used to do other power right sir so you will get a fifth power nobody knows what this thing is I'm even right now we have no idea what is not supposed to be can't actually these are completely new numbers in other words not a number we know not like a e like a pie they're polynomials that she becomes a new number not as they're earlier like I'm a number so how many number will ever learn from mathematics finally many not so many even we can't there's uncountable e the irrational number but interesting number we are obtained only finally many right so so you have to show this guy probably just a new number and now long new number people conjecture that for any other one you compute all the numbers should be completely unrelated to each other in other there's no polynomial term everyone two other okay so this is a window oil are done with 28 now to the city well I was 30 years old he thought something amazing he just take a product the composition of this thing see I really like this product right he takes decomposition of science now he's doing for 0x I don't believe this is the first formula I really don't believe this is the first discovered by Allah but you really use the Bible okay he taken to the product this a formula this product is not difficult to prove it's just there another way to write on the fundamental theorem arithmetic so every integer is has unique way to decompose the prime products so just to use the more analytical way to write it right is the equivalent no there's no any difference but what I use is he take a log of the highest both sides you take a lot zero of X this size you take expansion is saying you get this past stuff so we know by calculus you've a sum of integers your pattern is the power bigger than Y is convergent so this second guy is a 101 means the convergent so the only problem x equals one whereas because when this guy to infinity we study calculus we know this is this function exactly series we study in the calculus right well is a bigger than one is a convergent P equals or less than one a divergent so you driver just approved a sin and somehow this slice I did something wrong I have some few lines here is missing so so these were this infinity of some are reciprocal primes infinity of course short even many primes but this identity is much stronger than the standard model there is even in many primes that's what's first a proba Euclid the 300 ABC Oilwell theorem say that well not only even in many they actually called pretty fast pretty not not very fast right so because of the reciprocal is infinite these facts is really Mack the Zetas function to be part of number theory that's a studying is the birth of an even number theory pick otherwise I mean you say otherwise also the methods that are found to be the one of a trafficking number theory so this is a really crucial year in the table of elements of a number theory that that's a pretty amazing thing so it started with fun a sum of powers and here it is cover this stuff I am I really obsessed with with this data function and he found that it started there a function at is bigger than one it's not enough for him he really want to study Mecca is less than 1 the 1 trillion who wanna do he he took a difference of twice so this is the second guy he get all the net users what alternative serious because I went to calculus away no alternative Cyril has a much easier you criterion converges in me the only as a bigger than 0 right give it alternative a series of decreasing sequence the convergence is just if every chunk of the zero is convergent so that's what you are doing ok so you're not really find is that I found about this method okay now it's great it's bigger than 0 or it makes sense right so next time we're teach ask you computers as less than one you do that by this way you're convergent right so this guy is convergent right so before ok if I ask all have we are worried about it for I realize you don't worry about it you can compute by this way so I will re continue same and not you this method Allah knows how to get asymptotic formula let's go to one why let's go the one what this guy is one Mars hard frozen 1/3 that's exactly explain here Ln of 2 right let's go someone what do we have this is equal to zero but if we divide by a so go to zero liquor Ln number two that's kind of laboratory use at the time you know so this is this is he could asymptote your formula without any other tours then that's a nice yeah well I said where annually do you want to get a negative numbers I crazy you know you get a sum 1 + 2 + 3 connected numbers and I said well why not alternates you still probably divergent I put some power there what I prepared here this serious my radius the convergence equals 1 but I make s less than 1 if I'm lucky enough I calculated that take a limit there may be a current number well he actually computed there as an active integers often this identity so doesn't mean that if he carted 1 + 2 + 2 before I became negative 1/2 people say the physics are very brave mathematic we have to be proven track 2 + 3 can decode connective 112 he computer one part to sequester square become zero that's bit bizarre right then he computed the one plug to the Q + through the Q get a 120 yes the even power get zero or even power sound because the infinity always equals zero power load calculation let it be the strange but these facts actually will actually to the sum of their their the other numbers this actually has a relation with Darwin exactly because this is zero we don't know how to how others in okay I tell you how to do this thing it's not hard the older the work is really easy to see because one part to the cable at this end or not River they say at this guy right then a lot X here right put answer everywhere then where are you sick of this guy this guy is just the differential this guy many many times but into furniture your power more one but other side when you know why you would apply one there so you get this same but we can say that this guy or not like a geometric series so this cut now we know how to do this in well now what do we do we change a variable right if we do this thing we change X to the G to the T put it everything bad you got Pannonia number well this gives you another thing so all can be a boundary number so he computed the positive even numbers even integers and an active on a positive even D and the negative are integers he found there of the relation there so Allah Father that the formula I this former would approve what era was 32 years old okay so 26 April 1 transit 28 April 134 a sedative Row 1 then he got hit to something else okay so so this is the period just with the power of calculus nothing else your first a year caucus work can be used to fill out of things yeah right then what is the way if I started more mathematically you would do something more definitely so I will show you how the harmony analysis or analysis can be used to promote theorems so this come to dirichlet directly has a family-run he I mean directly the mathematician in in the Berlin he actually pretty good guy but he cannot pass his the language exam but he so he had a no PHED for a long time he has honorary PhD do not know periodicity so he had to have two jobs because I have new PhD he worked in the military and also a teacher and a local community college but he's lucky enough he married a very rich woman that's he he survived well you always find somewhere to do that right yeah so Richard Rambo Hodgson hunting go right so so he proven theorem is that any arithmetic progression than even many Prime's later on when cows retired of his position then Tok Tok his position so you know he was a really high regarded but at their language exams a language exam so I have no idea you hunk on some university one object language exam during class have no language that we could be this way so I understand them right so there's a lot endure mallet is much harder than Mandarin right so okay so here prove the theorem k prove the infinitely many primes in any arithmetic progression he proves them quantity as oil are doing but you're clear use something he needed use Fourier transform so that's something that you'll I didn't know that about it so you see if you knew theorem you with new facts so I just reveal a little bit of Fourier transform a furtherance of not everybody know before if you study in a college in if you don't study maths probably don't want it together if you study maths engineer maybe Fiske see me that you require otherwise I'd reveal little bit of Fourier for a transform it's a very simple form we know for periodic function so we know sine cosine then there the Fourier transform said that basically that's all the function you you have say the building block so any function can be right now as core especially decomposing Camille Radner as this function I use the Oilers function there and then this coefficient not difficult to do it just get this sense so basically every function it's a combination of sine cosine right I use their oil affirmation to do that because this way it don't really worry about okay this aid and so it is something there it's you be this you be a but I want to use a separate different version first is additive version so there's not pure define not be referencing the vanishing over all real numbers is square integrable it don't worry about the same that this this integral is almost like a sound the whole point says every functions almost like exponential function he also confirmed in humorous I also need this to started a multiplicative version but in bad value so if I only consider pharmacy from 0 to infinity I use this measure to multiplicative measure they also have the expansion this expansion is aesthetics when I have like a powerful machine yeah you see if you only study calculus there's some two most important family it once explanation family is the power function why was that two families yet because they are actually almost ascension power function addition term unification I mean exponent extradition communication power from the technical mode you get into modification right so that's the only thing of doing their relation is the Pyke's potential so this is a Fourier expansion for for function on it in front of 0 infinity but you use this as a measure right I'm a from here at UK it's very easy just to take a t2 there right see the log Z you get the back is not too big a difference but I just reminding you this size it's interesting but there is a humongous interesting thing for this one reason is you see this side estar to use the imaginary number so this is formula actually transform the real variable function becomes a variable function right so even this is your problem identical but this is giving you a mole more room to play right so this is the interchange between complex analysis to real analysis so I will use Sosuke discrete Fourier analysis tutor for analysis is that your time is continuous if some guy said time is discrete discrete Fourier analysis this actually very important subject you know you know you know in the real world so I just consider my function say it's just like 12 o'clock right and it's 12 just from I don't I don't continue everything just the 12 car that I discrete for analysis so every function and this this fan has said it has a for expansion the forest venture is by by this end because this stuff so this this is still sine cosine but it's a all of them it's a it's a Russian number and not only that I have no new k2 version - so this is this is like a real number then like zero infinity for this guy I have every formation our numbers prime - and I have this expansion so I call the characters right this side I call is phone analytical characters it's just some spectrum don't worry about that it's some kind of analog of the power function it right hand side but I don't want to write as I'm done right so I get this in okay so that's all it is satirically I want to use to creates a much smarter than I mean it's not really much smaller than Allah but you know something more than Allah then of course you if we started with a in there in there if a is congruent to if we started a congruent to mod n you can expand your Dirac function in a spectrum and motive creative word so that's what we're doing so this is something one study you put either IEP there you know you put it now you track this condition right PLA is a it is a Dirac function then you take a spectral decomposition don't worry about this number you get this stuff there any pretty clear what kind of tree the old inspire your I already knows it's it's infinite so you need only this you know other guy is finite right you prune some sum is infinite one of them you know the infinity other kind in the finite it's not difficult stuff ok so then he just copy what we are doing right into this guy I do this guy just exactly follow the Euler here's what this function there this is error function the top is the one he also has an Euler product right it's just a copy but but here's another obviously the purpose and I want to show this in the infinite he wants to show this in the finite but their purpose Sam okay okay I do not take the logarithm I keep on taking longer than two right well you're lucky that this Y is one of a pay he got a copy okay right well he I'm sure this is finite so you can assure this guy's is a finite are non zero has the opposite purpose now how can I show this guy the final purpose he went bad well I mean I see the some of these guys you know with the top in the middle Kari is hard addition is always easier well that's true right when we have elementary at this is a much easier than modification ok so he won't convert their multiplicative business to edit Edition business he then he uses their the Fourier expansion for additive group was additive group then this error function right at down at this guy Tom this guy but this fan this additive good means a fine a of n because Phi 1 o ends the power it's like a power function you get this guy so this is easy you put in some power here we get this stuff that's what we learn okay so clever is it together theoretically actually calculus number but it's tough he want to shoot this a finite he didn't show up I find I by by the modern waterway we take an estimate a blah blah blah he calculated everything he could a formula the to cut class number formula he Addison but this got nonzero he proved it here done easy just show you there with a little bit for analysis he can promote apply the theorem the oiler for example Inc was the for you get the oneplus one 31 5/5 you can PI before but probably this guy you come to pass some other method right like I like expansion something like that now I really come to another period is also very important their environmental period with the real end to end Raymond I of Karima with the crazy mathematician but they say that after earlier proof this summer reciprocal prime is finit Thank You I'll stay home he's trying to understand how the prime is is distributed then I calculate two three five seven everything in the big interval then he put like 20 pages I saw the 20 pages he put it there how many pram there and it ended to make a conjecture from zero to big X how many Prime's is really pulled by hand not how he does not have supercomputer no big data just come by hand so there remand he really want to prove the girls conjecture but he was a young student well the room and doing remake expert and the compressor variable everything so Reema forces in but well I mean there it's not a I consider they're just their first of the first thing want to do it data well it's not it's doesn't really make sense you already consider their the real value that should convince the Commons of value as well so you get oil up for that this little sense to start with oil on the rim and discover several extremely important property of this function use it for analysis the real numbers or written a Fourier analysis I mean they're directly to you the for analysis is it's slightly motive it's it's a is a slight modification Remis and on and I just use already no one right what are you doing you can start with the camera permission can't no family it's a very funny well why can't family the so important see look at the camera for me come on you right this way is the in fact the product of two functions these guys are what exponential function this guy is a meditative function so these two functions one is CUDA function additive word what occurred is function in the brief KT word this show you the gamma function actually tell you the relation between addition the modification so this is why karma function always appears in the mathematics the most important place because sometimes you think that you want to do you want to transfer modification to addition you interview a gamma function right okay so he's got my permission here he rides there are family are there many and transform of the set of functions so Salah family I mean they're mentioned is just a Fourier transform here at is away he get this guy this is familiar to you guys right I mean this is like a girl's distribution but it just sounded cute real from one to infinity it's not a terrible function something we know accepted everywhere means you put an end all the time even N equals one we are done it just put all in there right there if you use I mean in telomere repair a Fourier transform there are families just Fourier transform of set of functions but this guy is much easier I mean not much easier it just looks looks not so scared for right but this is son of NZ power there now he could rewrite is saying the sum from 1 infinity even it to one is a Poisson summation formula he even get at this very close formula but this matter is very funny F of T is a kills the sum of your distribution but everything is 180 decay so this is quite a very fast so this kind this kind of a Cynthia this is a polynomial function so this definitely is a whole know few continuation the complete plane not only that you also get a function equation so take care of member of continuation and from which in the equation he also tell you what where is there the pole right this is the what's the killer approved by his a trick but for for for for a man this is pretty much the cruciform idea why but this is really interesting so remain using the original Fourier transform not at the motivate for the Royal transform and use the gamma family was the inventor Paola give a new expression of this n next time but really just like Euler he only likes can sit analytical function as a polynomial well you see this is becomes I mean that the hugest thing a lot of to remodel stages the other thing a lot of time does elephant machine is only defined as a bigger than one the remnants of time is the complex variable function everywhere so one thing you want to attempted to do is that rod is function as a product in terms of roots right oh they key function away so so you use as like sine functions you write it as a plot in terms of roots under the product formula as some of the facts so there's a two different products to differencing whatever primes one apart the zeros so then here prayerfully now have to show that to conjecture that all these zeros must be in the real line right this is because okay if the girls context is true this must be true effect and remarks is equivalent to their generic form of these Prime's this is this is the formula this actually is a much more precise than cows cow see it's a little o of XD a little of this number really actually gonna have a plus epsilon reamer not only prove the conjecture also compute the first few hundred zeros don't know how the written my computer I mean you can't study calculus if I say estimate how many how many zeros in a lot of real lines then you know how to do it how do you estimate well if you're really good at computer every zeros usually very hard the slightly easier method that we will learn the calculus to see what the changing of the sign positive sign next I'm positive inner side it's just can you assign give you a zero at least one zero so that's what the roommate you guys are doing he sure that there is a lower bond of zeros by calculate how many change of zeros but hurry to the upper pond oh yeah for for complex variable you know how to cut upper bond because you have a cookie integral you take f of prime divided by F you take a contour integral right this in zero will tear your hole zero insight but as a Content this upper part is older older region right so so Rhema just do this to method to show that all zero should be there this is real nice ok so I'm at this stage I really got the twentieth century already right very fast ok so now there was the left I cannot guarantee and I I can make everybody accessible so I just to keep up general thing what's going on in this century after this injure is not easy okay first of all a general Reformation Shiva Ted is a form so you have annotation form the end of NS and their complex number but also neither require to have a product by or inner product but it is a pasty not necessarily needed to be one month possess but this is kind of polynomial this is good if to convert me from here is the ever but no yeah I just give me a bunch of polynomial put them together now expand eight now so this is not other humongous difficulty but it is not another additional additional requirement you should have a meromorphic continuation complex plan satisfy your function equation the two requirement together said make it almost impossible to finder verify it by hand how do you find it I mean the first guy if you trouble the first condition you drop secondary new I can do it it trouble the first kinda also I can do it right but to condition to care there is a harder so this make you the error phone which is a very rigid object it's almost impossible to that but this mathematical active rigid object otherwise it's too hard I mean everybody go home modify condition do they better come back it's a different answer this is a wide number theory is really difficult to study because probably is a problem you cannot go back to modify error phony about little bit so this definition error function but you are not gonna find an error value by hand you need to have some could take a theory to do that so how can I find a function let's go up at ARIMA what the remand doing give you it's just some modification product by this way rement of this way so real men discover your family by many in trance phone yeah you guys it would use a Fourier transform to get the error function Raymond just early lets us to do it about your metric method just do it this way so so this to properly give you your two different world why is Himani analysis over once the number theoretical world so this is exactly so this is a level inside the purely our resume attic reflect the true beauty of primes right inside the Pureland an attic edges the decomposition to the Mankato spectrum of this Gaussian right so we needed to do something for this to sense the two construction can be generalized to high dimensional situation with the Contin points algebra variety at this side and asperity conversation function that we use this is a passively not to constructing window right so this certainly so we call one side a more typical function outside optimal figure function so in 1960s level and the proposal program to connect the two constructions so large roughly speaking valence can you say that the more typical function a standard called the audio break automobile franchise in central assumption this approach can connected to different words of mathematics are resumes in harmony analysis so one the purely number theory you two oughta break way ssin you solve equations other guys I don't really care they say it's just a bunch of operator laplacian operate these Opera House or readjust computer spectrum these operators the other side that use a Hadouken or solder metal you know this idea or you can only use audio geometries it's just like two things okay so we're giving you in the left of my lecture I just give you a few examples illustrate so how another word and all a resume attica water I connected the first exam I want to do is called Ramanujan twelve function so this function actually is related to the tube I don't know this okay the point that doesn't work anymore okay so it relates the two they're the Gaussian distribution either better the flat girls division taken the 24th power you'll get is a path okay so they call error function does I give you the Ramanujan's the Indian genius Ramanujan I see neither 1917 samsara that he conjectured approved the later on this function if I take the variable little bit are we get I guess I lost the battery okay so if I change the queue if I'd if I replace Q PI e to the 2 pi Z this is even the product with the final so-called modular form or we're 12 so what do they mean is that that is a lot of function equations right so this is the same he show that and of course the his pharmacy is the one of the many functions the matter from Wed 12 you can replace the 12 by any number like 10 five eight whatever right but then in nineteen thirties the German mathematician hacker he started systematically not only delegate at all into all these functions and study is for expansion then he showed that if FX is has a holomorphic ur can show that each of the error value had form of a continuation of function equation that's one requirement error function secondly Republicans to the amazing thing featured he noticed some operate correctly corpus he found if this function sometimes I a confirmation for try co-operators you have owner product the second trait is some kind of operate like a like a laplacian is but in the p-adic world diplomatically flossing everything of that way right so in other word if a function is a spectrum further for this this pru laplacian for every p I have an earlier product so so they're heck is Method TV you generate and produce many many these air functions but there's a one set it back because he'll camisa de is it depends the spectral decomposition we can count how many spectrum inside but nobody knows how to write every spectrum out write it very very few you I can say I my spectrum increase like this but tell me can write down by hand sorry I don't know how to color right now I very few some of some set of functions some some nickel raw data but for general one there's no way the passive everything we're right on is that you had a Sierra one nature for high Chi there's no way to write them so anyway so there in 1971 telling prove that all these holomorphic module from a motivic so prove one direction all over the lowlands conjecture right so one constructed by heck actually also come from algebra geometry so this harmony analysis sky the same width on purely by her money analysis actually has a correspondence in algebra geometry then he proved the famous pyramid Ramanujan conjecture all the coefficients are bonded by peel the cameras over 2 that's also amazing because this this coefficient come from analysis when we have no even right now I have nowhere to prove this in quality pure deep analysis this is a pure biology arbitrary okay so as I said that it's very difficult to write on I mean there so now I talk about other direction other direction are purely from and construct from matific advantage on analysis so the reasons waiter for more typical one is star to the elliptic curve elliptic curve is just a bunch of the equation y square X cube plus X plus B these are very many right this is very great already many you write the question there you form the Euler product the oil product is always a polynomial three times there's AP EFMP but don't worry for almost all pee this epson p0 the S&P well find many peanuts in PS 0 AP plus minus 1 0 it's so in other words some simple effects for otherwise this this this epson p is always a 1 and it's number in the middle of one must a B plus y is a number of solutions so this is fairly easy to calculate the reason is it give you an equation usually if we ask you to solve is very hard but even solving I find that the fewer mark P right you can use a computer calculate as many as possible so this is the so this error function actually is the constructible object is not other existent object so in 1994 why is a problem modularity for an every lifted curve come from a modular form now he proved first for semi-stable it occurred but for modularity proper 2001 by his students his some collaborators and this is I think this is really a time where the people out wow there's a lot of programs is really used for subject why because can be used to prove the framelit theorem so as a consequence approved firm law theorem there's no integer ABC and we're in picket that's three is employed to build and you pass it then so I think it's that time there the average a formula the application numbers here get it at a double increase everybody won't come in but NASA is the proof another theme like that okay so I don't have time I think my time almost the finish there's a many I discuss the Riemann hypothesis but it's just cause Allah and the program there's some other subjects which I more involved a special value of the material function so you should relate to the PhD conjecture Taylor conjecture list the to Brasilia the the result is the girls idea in the beginning relates a special value of error function to three elliptic curves there another one I think it's a reason to work up by jinjo and at wrong way it's dozen 15 the published in the I guess said this year the children at this sense so this to work probably most exciting worker and the post special values of modificator function my title is past and future I've missed the present the present and there's no way I can give a lecture a public lecture you probably need a one-year what sharp and discus in for futures also ease these three lines so the river of the land and the progress special value are zeros are three major target number theory 26 21st century isn't my opinion okay so for young people you want to come in this is probably something you're one one of them you want to study so usually study whatever but the truth is much beyond this day then the calculus how many analysis the thing involve right now is the fairy order by geometry periodic analysis it's totally different in it's always a true you're starting more the phone number of mathematics you can solve more problems right so there are few champions as the people should have study more than me right what I do know is this kind of thing ok thank you very much people is probabilities day statistics charisma the statistics we Jen told me that it for the graduate application almost everybody wanted to how many people the rank distribution everything like that and other than I'm still we're still classical PST steer rest rate and rank 0 rank one for function field of course I said the region and two regions the work give us the infinite even money imagination but just the imagination will have a sofa have no idea what what could the faucet is a comment okay the Riemann zeta-function I read a bridge to more people they said that the Riemann signifying can be considered as a masher for the prime number or crimeans okay because all problems or prime numbers of the set of all primes is infinite we could neither have our country back over there but the free sport prime of the over 1 over P okay - then - so paint it now you're some oak hill for you okay yeah you'll get okay no this for okay forever heat is you okay hello put 0 1 over that one will it get a weight you get a sum summation and that one is the total mass of the positive integers no okay oh well okay for the a or functional okay where to do something to make nests okay you have the whole a or function had some okay night like the total mass or before or what's your magic or interpretation guys yeah you have this you might mean to presume that's for some coefficient or some other things okay whether the error function itself can be as a generalized module or for some certain space okay I speak of our outreach externally request slightly more so for example there the Taylor theorem I mentioned a blackboard right so this for each a you have you have a residue class that's expected interpretation for the remains a function okay judgments that you see the series okay you mean to give any series you can do you know gets here okay well here's a finite is it okay and that one you give them night-night oh okay sorry pollution whatever becomes the fallout of functions every Majella function you become to Bishop rook okay I'm Madi so that kind of interpretation for the that's for for for women's ever functional you know whether it's some kind of only when you have a pole you don't have a boy don't have anything right if you're already converge and what it in approve so so you're talking about the best units rumors are finds only deformation okay there's a big conjecture it's a it's a PI R 10 and nylons it's here so only be function has a pole the room is that a function so that's so unfortunately only that's only a sin there's no other error function is a holomorphic so our x equals one is a finite number you don't have we don't have the phenomena you just say but there fairness of error function means what it means there this angle pretty smooth the only one direction even a distributed so there there what he said is really the fundamental theorem primes right how many prime this was an ex keep a formula you also can do slightly more complicated there's a so-called Chiapas how the density theorem then of for for distribution of the fruit of the primes in number field for modular form which is not from the Murphy laser saturated a conjecture the probe by on Richard held a few years ago nothing reflects is the same you are doing so you're talking about statistics of primes right boylar proof of the good result every other year is getting Pena here I see the bus our problem here we're gonna tenure and it's that so so so you're you can against him because he probably too many papers so it's hard to yeah I only wrote so many peppers it took 50 years after his death to publish whole he's a colleague to work so he's a song the guano son continued to submit a paper for publication it took us 50 more years after he did so then yeah so he could be problem in modern society this guy did were too many people who knows which one was more important than other guy yeah and other hand remain only republic only do a few papers maybe nine nine nine six papers and and I guess in in today into the other world in pure mathematics department if you have a quest to dance probably every students has one way other way is a walker is completely invented up I remain not as I related it really invented Romania geometry of causes the one thing I mean went to number theory to have been a variety I've been a variety is not a buyer boy is it by Riemann right he is really the guy to the high dimension above only the one dimension right so so dancer so it's it's a anyway I mean they're all your life is just amazing mathematician you know it's just I could not believe a well I see how how powerful he can use the calculus even the calculus the first here we learn in Riemann hand is already burning algebra but will also learn algebraic geometry and I just cannot remember some resulting community algebra I like to could you please tell me some some of your experience about learning but a memory problem okay I tell you what this is the interesting phenomena yeah older as a mathematician will wear that shorter shorter memory than a physicist not only to you everybody so what do we do is always it like that every day I have a pen of a I mean the question would not we're not a required memory we're requiring repeating so every day of a pen of a of a path I can 99 percent of time you are recalculate the interviewers so what do we hope it's a following times because I forget what I'm doing last time this time recalculating maybe our to something different so there's a different thing if something was a lot so I think if I forget up some same Minzy are really good a mathematician because you started recalculate the whole thing you want to do is take a pan take a pad recalculate don't really initial each a because we prove it like I like house there reproves you know many many times right before it's a time they have a saying they're gonna redo it so the what do you say is no problem I just forgetting everything I want to get it ordered I mean except my name I still remember I don't remember anything yeah yeah last night we have a comp we have yeah well he hit sir he call his wife he asked his wife they pick up something in the refrigerator do you know what he said it's a Tom Stier Louis yeah I guess whether Natalia had chosen some poor a p-channel do it Louis yeah p-channel even the Louis yeah because we cannot forget that word yeah so it's a that's not a problem the problem is that you have to you usually you used to have pain a path to the calculation something I forget you start something remember to deduce it right in two hours two hours does the same and you hope this time acquainted use your formula which is a different than the one given maybe formula the same but expression the different or expression the same but approve is a different but that's something you learn for yourself your apt question so suppose somebody as a student student yeah undergraduate student who want to do any of this Raymond landlines of series what's your situation where where where does he need to be good and one of this he doesn't need to be good for all of them he need either be could have money analysis or could in audio bruh I don't say because once he could have one thing you can get in the field then I have time to the other thing right but if you could proposal ins that's impossible right so McCarroll how much the thing we evolved at sir there's a representation theory there is a number theory there's audio bridge on the tree right representations here already involved whole analysis inside I think I need to be good or the one of them was it one of them you talk to teacher to just say well well since you know this saying that give you a problem you started work once you work on this a problem then you explore by yourself more things now who knows what's going on because at the end I supposed to know much more than you at the buzzer but that's not because they're so violent no way to teach you what the children because I don't know what's going on I don't know what I'm asking to know always know much more than me but when they come to my office they know much less less than game you put in two or three years they know much more than me so every time I don't know anything I tried to call it wrong way say I call the other checker that's it right there's a good use of students right okay thank you very much and for the wonderful talk and interesting exchange with our audience okay thank you grandma

Channel: HKUST Jockey Club Institute for Advanced Study

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Length: 74min 18sec (4458 seconds)

Published: Mon Mar 05 2018