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1/3 The pretentious approach to analytic number theory

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so I want to talk to you about the pretentious approach to analytic number theory which founder Roger might have been developing over the last 5 years and I some of you work at the summer school we have some years ago and have access to the book that never appeared on the subject so for those of you who don't know much about it let me explain what this is all about so approach the general approach analytic number theory stems back to romance memoir from 1859 and involves studying the distributions 0 reinstate functions and as we know been very profitable approach to health functions and naturally arising questions and respond to it for a long time and yet many firms have been proved without zeros of functions and to some extent the the community has said well those approaches are ad hoc look down upon little but the victory these reflect elementary work championed by pool and ocean many of agreed firms analytic number theory especially about the distribution prime numbers come from so-called ad-hoc approaches so a few years ago I Sobhraj and I have been using certain techniques that we call pretentious and I'll try to explain what pretentious means in these lectures 2 prove theorems on open questions to try and try and develop the subjects and we we were interested in water we saw become interested perhaps and what established their arms could be proved by his pretentious methods and we attended a lecture by Henrik Avandia and Princeton 5 years ago when he announced approved by himself and Friedlander of Linux there that in every irrespective progression where they can be Prime's as a relatively small price and the many answers proofs avoided many of the difficulties about the Israel functions that all previous proofs and our values and freedoms proved avoided the steep ideas about 0 0 functions that had previously being part of a very difficult proves it Linux experiment the the proven Friedland ran its is not easy itself but in technically avoids these trends difficulties and if you like is an elementary proof albeit a very hard elementary proved so it gave us courage to say Well maybe these techniques we can work on could prove everything in classical analytic number theory so we send are going 1st goal was maybe we should trampoline experiments cells than we did before we'd surprise ourselves that we have some ideas for technique and I worked and we weekend pretty sure prove limits much of much easier than what existed at that time in literature and so on we wrote up if you like what we could do in proving the classical results of what you might find particularly in Davenport and then in the area's lots of books and 2 from we were able to accomplish quite a bit but there is some clear flaws in what we had most importantly was that we couldn't get a good error in the prime numbers so I'll remind you what we know about prime numbers Thurmond moment but the number Prime's up to expertise and like Exelon acts and we couldn't we could achieve a narrative that little of the effects of log X and X logics some power but we couldn't even get a power to be too so we unable to prove that number president-elect was Exelon ex because big of acts of logs squared Exxon methods failed us on that so that was 1 big flaw in what we were doing it and that 4 has a knock-on effect because if you try and prove a Bulgaria Vinegrad fearing which is central to many of the developments analytic number theory then you need to be able to win the prime number theory the prime number from 1st under progressions by an arbitrary powerful objects Maritime of logics because X overloaded so 8 that's what you need the 2nd thing was that that the subject and I'll explain all the centers around what school Hollis is there and the proof of Hollis is there or Hollis is proof is a little opaque little hard to really understand the motivation behind it and after Hollis there being many improvements to His results and that we had a paper improving his results but all of them started enforcing fundamental construction that was somewhat opaque and this meant lot of proves that we have really worked but there rather difficult and some why they worked in a sort of a larger Saturday now these I would regard as what with the 2 main flaws in in book we results 3 years ago and what's happened in the meantime is that those those flaws have both been removed and not by us but won by Dmitri Cook offices here do showed us how to improve the prime number theory and not only those strong Russian prime number theory but as strong as can be done by classical methods and and more recently Adam Hopper came on new proof of police's there and that no it's not incredibly easy but it allows 1 to serve appreciate what's going on and makes it much easier to use the techniques so in both cases but we've tried to further develop the theory developed what could block was that Hopper did and so that's what explained to him Of these developments that now means that we feel we have a strong theory that rivals the classical theory and while there should embarrassed Samaranch thing will get the book cover 9 months but now we got to work on it and get other people here I know who would love to see it out so that can use the ideas so I it will take some right because there's a lot of things are doing a man of people like Maxine keep on proving consequences that 1 would like to use it makes it difficult for it's a book that will prove my spirits and to get it finished book anyway so let me let me we change the title because both of these new proof I come down to so doing something a little bit surprising kind of cordless Tehran's formula revisited that's really going to my title for the 3 books and in the talks what I hope to do if I have time is proved your show you how the proof for the prime number 13 goes the strongest form grove Garbo form of gives you the new proof analysis on external power system is and hopefully prove limits there in a way that even understands so well when using thing Brady the case so 1 of wanted to start with the classical proved prime number theorem and I'm going to assume that you know it already no 1 by 1 month says 1 knows approves a prime numbers the to prevention just write it out because although the ideas very elegant really are yes there's a lot of steps in half with a lot of estimates work in altering the prior approval prime number 3 months adjusting to quickly go through 20 minutes approved will and some of the notions of going to the usual proves a 3 will come into what we're doing so if you start off with a basic definitions so land and it's just a convenient way to that we what we want to do is count primes and so we want wait 0 offenders are prime fast convenient count prime powers for technical reasons and that if any is a prime how we can see this is a mangled function school and it's 6 lots of 4 minutes so to do the prime number there instead of counting the primes we count them with this weight and instead of Exelon x expected with some merit already the integral the key of the lottery we expect the maintenance acts and then the question is what Keraterm and Selinda just give to maybe straight so we get bigger elects the half loved square decked if we assume the Riemann hypothesis going into the room hypothesis little later today and so on the best result 9 years due to but if problem than a grant and it's log annexed to the three-fifths blog next to the 1 so this is unconditional and that's the best result known to the constants and then from the 1 you see in the textbooks if you put through the proof would be something like this OK so will will more or less give all the details of the proof of that today and I will indicate where you have to get this thing here is the home and said that OK
so let's just remember Peron's formula so this is a beautiful idea for but recognizing an inequality monks integers and very simply so this is really part of the sea is greater than 0 and something happens in between anyway the important thing is that well it's the average and the important thing is that we have a way of recognizing inequalities for real numbers using this integral and and hopefully played it before so you prove just standard way of pulling the counsel to the right or left and you either get the poet S equals 0 which leads you to listen the value under control he added controls small all you put it to the right in this case and the snow Paul the value they added Contos small so it's not very useful to have an integral going all the way to infinity so what once is truncate this at some height and then 1 kind of workout inheritance alleged which looks like white to the sea this is all define this for instance in Davenport's work this is why is not equal to 1 minute flight schools 1 you get some 130 this year yeah so the point is that while the you can work out the point at which you can truncate and not have to worry about their time I'm not going to get that much I'm thinking about that it is going to have to trust me there is a point but again I'm sure you've played a separate from the prime numbers from that's the case so what is the key application of crumbs formula it's too remind guests yeah we are I wish to look at the summer Paul and up to excellent just assume for we have always had a problem with white equals 1 that it's a bit weird so that he makes is not an integer so I can't be correct and not have to write extract then the way we try and some 80 sequence of interesting numbers and up to access is the right some overall integers times 1 gift N is less than or equal to X's and 0 fan is grateful next but added rewrite that his x overran is greater than 1 forex over and it's less than 1 OK so the idea is then you can just plug in what we've got up there can we take Y equals X and the plug-in why calls Axa ran without incident Solomon an infinite integral if I do think the top so the point final flexibility with the CIA's we can put it far enough to the right we can make things absolutely convergent and so the order of integral scroll and summation and so we end up with 1 over to Itajai let's have a look what's inside the integral so I've got a hand and I plug for here why cause utterances XPS over into the S so the positive interesting varies with is this and the rest is actually yes arresting K and a what we can replace would just call this thing a that and so this is from the application of this formula that is most useful in number theory so In order to understand the sum of something arithmetically fail to Ex all we need to do is understand the integral all over their face serious time something on the internet lines so you could argue we taken a nice finer discrete come problem made into a hot analysis problem but sometimes that's useful OK so are some of the most famous applications will be for the and we want to work for land ends and all you have to do to be but if you go to real part illustrated the 1 they knew right has a dress is the sum of 1 ever into the S you right it is the or the product of the primes them where you're over in the domain after convergence you can do things like logarithmic derivatives and this is what comes out when you do the calculation carefully so now we can just apply the formula land and we find that the sound enough to acts all land and equals about 1 of the 2 hours time the integral well let me just could at old 1 go just believing that you can some suitable to -minus later Prime Minister of Israel the arrest and actual just say what the right here at home when you do all that so again this is all standard dilution B plus log X because those careless what happens when European we took 8 so what's the game now is we have the integral of a rather hairy function and we have it the writer of 1 because we're doing on manipulations where you can now do manipulations without too much worry where things absent from Virgin and again we want like complex analysis so what we're going to do so this is going to pull the left and the hope because we got was integral to hide the years 12 treatment throughout this and so the hope is that these into GM mice and small the value of the beans growers into grounds and then everything will be picked up by by and how she Stirum which is the residues of the polls to be the inside so will talk a little bit about the whether or not it's easy to proof that b is grown these lines of small but in a minute but let's just think about the Poles Soviet aggression toward other polls Paulson but well rooms a dysfunctional Martini get into this but as I hope you know his very Moffatt could score a pool of water 1 equals 1 it's the only poll so equals ones HOS looks like 1 arrest minus 1 plus gamma plus when I went past tends toward someone In the expansion physician and so that's the only poll now OK why my saying that lets some of 1 of the polls listing likes the S has no polls 1 arrests this apologists equals 0 that's pretty benign will think about that in a minute Zeta prime privatization we love and have a poll of Zeta prime or zeros the polls estate of crime and the same as the polls later because state with minimal effect and in the series is a farewell advisers Zeta so what we have we have and the poll s equals 1 and so the politesse equals 1 gives us what well if loafers a poll of old wanted as equals 1 Zeta primary was Zeta has a poll aboard 1 of equals 1 residue minus 1 so that minus 1 times
minds want 1 plug-in S equals 1 2 here we get extra civil 1 so that gives us a residue acts In a sequel 0 we have the residue minus the 2 prime 0 realization 0 and from otherwise we've got S equals wrote where Sager wrote equals 0 and those are the possible polls case of this these 2 we can handle easily this requires more forward so this is the difficulty in the main difficulty agreements approach is understanding these that's that's was difficult and it's still pretty mysterious at understands act so what happens if you have a poll of order and here when we take as primal as we get em over its minus wrote In and A-minus a reference for a 51 right the former we act as if you got em separate zeros which makes it beautiful to write that the former becomes X minus the sum over these rows of tanks the row over road counting and times of the 0 board RAM -minus data prime 0 overstated the connection proved this exact formula collects ego to infinity but what's more convenient for us is null and the infinity and we get the same errors other cases law standard you know this well very benefits to the schools it's worth going over Everybody has revealed that so there there are losses zeros perhaps a rise of 1 and was the bunch we'll talk about that critical stripped some along relaxes and back also I don't know maybe have a little bit careful depending on how far back at had a comfortable with a reunified thinks spoken so now the question is can we make this argument work now you have to think things through like for instance when trampled Contra across here 1 if you happen to bump into a 0 therein zeta function as you go then you're going ahead a Paul this is not going to be so easy the just bound the incident so what a good idea seems to me is to take the 3rd 2 0 7 cent function is you take your Contos sneak away halfway in between now it's that easy to do yeah it's easy to do because what you can prove it takes someone is the number of zeroes up the the is constancy velocity the which means kind of an average gap between zeros of 1 of the largest 2 pliable in fact so someone is going to be such a gap around them and you just go halfway between 2 zeros and that allows you to avoid the problem of stepping on a pole so when you do that you can get some idea of the size of Satan primers agent and so on well the come short In this kind region several part between minus 1 you can have something like miners say to Prime Minister Jose to but here's dominated by Willis not perhaps not surprising that prime of his 8th there should be something that the sum of 1 embarrassments yeah that's what appears invulnerable and so this is dominated by the -minus rose at a closing and was the at the Rose this is some of the series wrote such as minor stroke is less than 1 and you can prove plus some of so with this sort of estimate you can work with Zeta harmonization and then not get into that too much so what I want to do is just focus on the formula here and see what we can deduce from it so Oh some course will subside that so we want to bounce we wanted found the error it's my effects and the main thing here is the sum of X over Rome will be nice is to be clever enough to find some cancelations between the terms for different zeros unfortunate angry enough to do that to the best we can still do something like this some these rows of constant value of X the roadmap barely rope plus well it's a constantly visit Prime 0 0 so we just need to to worry about this guy and I guess I'll take tea in arranged just so that have a loss and 5 things as we go along now the reason write this down simply just to know that the absolute value it's the road a surreal real Potter Rome and its if roasting of society from forget the actually IT and has value 1 using the fact that the number of zeros at the he is like the loyalty and also 1 can show their no zeros of reinstate function very closely and to the Axis 1 can show that the sum of the zeros up behind the His lesson long squared partial summation and so what we end up with abound on this term which is like X the maximum part a real In this range rose socially important match reporter rose less than time squared the plus a 2nd this is the key calculation in getting airtime the prime number theorem and so on well you can see very clearly that what 3 important is how big the zeros kept in this box what it's how big the real positive 0 scared now but will be from the is that from all the zeros switches we said the uh 0 somehow find that billions of them have now been calculated the on the half-light so if we believe the conjecture that billions there that always serious from offline so it's all who is falls road satisfy real part of equals a half all lesson refer half because of some zeros further to the left then while here we got makes the half time square carefully picked 2 years Texans will also be extra half unfolds square so then take the calls and texts and we get nearer-term what what we said over here it's half time spoke on the rehab hypothesis under the assumption that all of the zeros lie on the Hofland or to the left so we look at it and bed is what happens with regions that have just said do it rather than on previous around so what will eventually prove something that real part of growing this box if the Majorie part of the role is less than or equal to to teach them the real party rose less than 1 minus some constant last 2 and if we can do that and we just plug it in but just forget logs so these are the main turns up to the logs and then if we want to Max minimizes rather the the way to do it is to the quake the ecstasy of loyalty with the exit the that'll require us to take something like 2 years but exponential-of rude log acts and then when we played badly and will get the standard error on gets in the textbooks and then this error term requires a better 0 free region so going somewhat further of Love OK so what matters to me pruning prime numbers all is a free region and from let's just talk a little bit about how 1 goes about this so the year the 1st step if you like is to show that you can't get zeros on the 1 line and then to come in a little bit so the Standard proof if you like shall we say this so let's suppose the Zeta wanted he was 0 and then we have something like what the the product of 1 minus 1 of key the 1 to minus 1 0 and what is this
kind of telling us you've got to someone of people wait and you've got this piece IT and arguably is telling us something about the way the PTI IGC .period if this was to be the case so if I was to take say the absolute value also known as the wounded so I guess what I want to say is that yes the PTI team predominantly pointed toward 1 of the positive real direction that you might expect this something that 1 plus 1 of 1 minus 1 of epitomize once it would go to infinity so there is a way in which 1 thinks that he the points predominantly in the other direction that's the way we forced it to go to 0 so that averaged out the dissident .period direction you might expect this to sort of be a constant a bit like result functioned relations in the previous lecture but Was right in this form so this is approximately 1 plus Peter -minus overpaid so yes if the pink the ITS look like they're pointing towards the positive direction primarily than these outside typically be given 1 the case said the overall product my differences the but the overall quality is going to 0 so expect upon predominantly point in a negative direction predominantly is a very vague term that anemic less faith in a minute and then so this is actually were have Davenport talks about things before he goes from attends proved if that's the case the PTI genes tend to point and negative real direction than the pizzas to IGC Ophuls the squad tens of points in the direction minus 1 squared but then the kind tells us that we should expect Zeta want us to IT to have called for that diverge "quotation mark infinity but this had fairly simple considerations rings at a function tells us in polls equals 1 so this is sort of a standard heuristics to finish off the proof the prime number theory and this is proved nicely using a clever identity of Montana In probably treatments you read but this is if you look at both the level of 8 % at essentially trying to capture this idea in the group's Izmir 10 came later and you have the proof of the co-signed identity that came afterward so what I want to do is go back to this original heuristic and try and understand that little better so To do that I'm going to make the definition is so use them off sport so 4 coefficients and greater than 0 mm 0 let defines the distance between you functions and of funds into motion rousing but from the we expect so this is not exactly distance function maybe I should just cut to the chase of the simplest example which would be if we take a hand to be 1 of appear fan is a prime and 0 otherwise the whole thing works In some general state and what's the idea going to be the idea is that we have to multiplicative functions Hassan synergy both inside the unit circle and this is some sort of measure of how close Stephan G. onto each other so that thingy were to be equal and also to lie on the unit circle than S the conduit would be 1 everywhere and so 1 minus 1 would be 0 in this so-called distance function 0 is not quite a distance function because of ethical G and were inside the unit circle in this wouldn't actually equals 0 but the something good about this definition is that it satisfies the triangle inequality and that's what's really relevant to us is that it Will several forms a triangle inequality can you just give a traditional 1 so it satisfies I have a preprint on the archive quite recently which is for the celebration of 100 100 250 of them listed the math Association America transcribed some these ideas and a competition to find the best proof of this so we don't like approved for much of 3 entries so far 1 of which is very good anyway and have a look that if you like elementary entities on the archive is all answered yes thank you we're going to the middle and only OK also .period also lists in the context that so the loss of function as is you can't think the right access the some of the prime powers also but 1 M P 3 and signal times please minus I am playing here I'm putting Aceh signaled "quotation mark thing so if I look at long positions Sigma should and so just when considers natural quantity then that's the same as we said before we take the absolute value something it is something real part upstairs it's a real part of the log 1 of his aides agency Zeta asked and then if we just use this formula what do we get 12 S insignia have this part was saying so we get some of the prime powers of 1 of the end it's the end signal and then for the Zetas Sigma this is the is 0 so this is just 1 after the other part we have now if you look at our definition of their looks pretty similar and if you like this is comparing with appropriate set of weights this is the end of the distance between 1 and on to the IT always instruments yeah so the final inequality will give us simply that well you can write
that has and then just by the definition of the SG prime I can write rewrite the 2nd 1 was just divide up by the same functional unit circle and Varitek just write about that so would this is equal to 2 times the distance between 1 and 3 I the this is what I think so 1 into vessel simply square this so I get myself in here substitution and then we get back will almost ,comma slight fail we almost get back from buttons and call next you get this thing but the Kaesong doing here's some squaring so this is D squared over there is less than 4 finds that the square and substituting from the line about no sooner Poseidon signal for when the smoke clears this is what you get taking expenditure so Mitterrand's Hodges remind you of ,comma 10 works in a minute but imitating a plus for them so this wasn't quite what we wanted but actually that's not very hard to report repair because I could go through the same thing again and so on In order to get a plus for that I could get the inequality for purposes of taking away from their functions here so twice so exactly the same argument assuming the square the minus sign really matter and when you work back but a minus sign in here fell for a dress up here and we
get a plus for Kagan just trust me that the other works to verify cells but so this distance functions somewhat flexible tool to create inequalities and then Mitterrand's famous result famous proved says that if you had a 0 here as you go at once society is to come into 1 from the right this because it's a it's an analytic function this is gonna look like Sigma minus 1 to some integer power the fall so that the signal Midas funds for this is what I was sick minus minus 1 to the 3 30 year we Sigma minus lease signal minus 1 the only way this whole thing could be greater than it's a wonder signal goes 1 is this compensates by looking at 1 I was Sigma minus 1 but then we have a poll here which is impossible that attends beautiful argument to finish off the proof this is the 1st wedding between the heuristic but everybody used and the eventual prefer we can consider that and not get into everything I was hoping to get into and so on let me however jumped 4 1 to close so let's go back to to the beginning of this proof the stock in the middle or answer it was just a looks at my work is lighter than sounds what would you OK so going back we we had 5 x years of the interview whenever to apply the integral and from Siemens signed assume my authority the plus of -minus a surprise at a very fast XPS arrest the arsonists we said in order to make this work we had seen the right 1 so see greater than 1 plus Samaritan worry that much of their and the tactic of Raymond so now we're going to get in into pretentiousness before tactic of Raymond is the Tigers this contorted runs of 1 and you shove it back to the left residues it's all very elegant and beautiful some technical issues need to be resolved but but wonderful so the new approaches 1 we just look at this and try and 1 white 1 do it's called for integration move account of how big is suspect OK so I want 1 so the step that's going to make life easier is that when we're trying to count the primes up to acts the technique we use doesn't use Prime's bigger than yeah we went looking for soap and this slightly changed then regret this later but in the next Federal have different meanings solicitation but safe excess will just be the disarmed 1 and the S and P divides and implies he is less than reek of his so this is a nice finite thing it has a lot of product so in the argument we use it was no reason not to use this thing right because it'll just give you the front of printouts all the primes up to expand the printer so will work with this is the reason for that 1 2nd and now let's just take absolute values here and see what we've got so what's the absolute value of the right hand side so this is less than or equal to take counterbalance to get 1 of its some sort of integral but not change so do some of limits in a minute but managers say prime of C climate change from let we ride the SSC plus violation the so goes from minus the plus the days primary the Zeta From X the city when we take after resurrects the seat IT rejects the sea and the denominator we get something like C +plus stamps value the deal something along those lines the accounts give it to him so the ecstasy is a little bit of a worry because that we know that the right size of this is about X-ray so the ecstasies amid worries about as a typical and these Contos let suspects see a tiny bit bigger than 1 so that access to the sea is conveniently some well-known constant time x OK so this is less than less than well 1st ways we can play this game we got was axed constants worry about we've got some integral Zaitsev has a very complicated vaporization this just take the maximum of it data its and then summoned to roll that looks more sea is very close to 1 to this looks something like take so how big are these 2 gentlest one's easy this is just like a log need to lobby for giving How big is this guy well I'm really going to be very crude and say I wasn't the prime of his data looks like the sum of land that but it at the summit land over the sea you have to get the IT and will do this friend to acts not 100 percent correctly there should be a few more times but monopoly importance the CIA's is more or less 1 so certainly less than land ran over and up to acts so it's less than longer OK so by doing something completely stupid which is just a capsule values How would you we lost the right answer is X plus an irritant and we've lost biologics analog TNT which used to be excellent half before we might be a bit more judicious now about the choice of the impact will take Teterboro power blog so I should be careful assesses them not say that he might well be the more complicated than that but we've only lost by some powers to block so that's surprisingly good for such a crude argument so another approach to patchwork and Prime Minister Minister Lester stranded little less crude here and see where we go OK so I'm so confused his board on crude hitting targets that once that it'll stay down for
OK so we gonna argument and we haven't done any analysis at all so know we're integrating the right to the one-line not straying into the critical stress so the 1st question we should ask is if we've got a function which say states this assumes a friend his data is a fairly benign function which actually is whatever is essentially benign I'm actually we certainly understand 101 understand is we know the right answers extra we know there must be some cancelations interval so where does it come from it was an obvious place it comes from we'll think about the terms in the time that we really understand well size the 1 arrested me understand well actually and what happens to the integral the as we go through short interval well intellectually as asses Exley as blogger but I'm more importantly ready for us is if we go through From S 0 2 S 0 plus what to hire log access To I'm tired Logan and so we go over short integral show part of the integral up the vertical axis then that differences 0 the case very short intervals like 1 of longer acts the Exley as part disappeared so there's a win that we just have to learn how to exploit it and we exploit the cancelation thanks to thanks the fast so what's your obvious way for exploit that will integration by parts allows the 1st integrate the Exley s and then hopefully the rest wanted to nasty when we different OK so if I was working with an mandate to switch back to another location from which is over there so if I'm integrating they have asked Exley as the rest of the array of SSA to prime since then what's the idea now is in wintery by so it'll be made to the U.S. arrests the integral actually in which another Woods's XPS ever-larger acts and interesting parts and then we've got my the 2 other terms of minus the interval of a prime minister a longer the arrests plus the integral all of Over the next 3 years of Esquire the logbooks again but limits a somewhere about limits what's going to happen because of this circulating around so much as we can base in consider that to be 0 and with a full view at infinity that's certainly an identity and action to study this is an identity let's just take this and go back over here so we know this represents some a N N up to act what happens in this 1st time here what is a prime what is a derivative of their faces well this is the sum of a handover into BSO minus a prime Is this summer a N during the past so this firsthand corresponds to the son of a N Logan over log acts well I think you can get the 2nd part represents an emphatic system standard tricked using primes formula OK so this part actually is easily bounded now assume all the a and Celesta to 1 which not quite sure no problem but is almost ruined the average of the lender rinses but by 1 them when we do we can just taken after abound this West 1 here so the someone slogan logic and this thing is less than excellent objects of innards justice so was 1 just 1 in here then this would be X minus blogger factorial little longer acts and most of formula but the main terms acts of a lot so the meat of this integral here the thing we this is small and we win by 1 Log acts anyway the meter maid lives in here so let's just examine water we get out of this integral as opposed to before so before I was when I was integrating this thing and taking up the values I had survived the maximum I interval very arrests times X secure again and the U.S. is 1 was 1 of the log please cite the and thinking of an integral from minus the 2 plus the the Times slot the Tunis-based you 1 ahead of their own With Zeta and now what the White House will have something a little different when I make the same so this is the bounds but I get they captured values in this integral now I've integrated by Parts 1 time and I try to do the same thing what is it I get out of this thing 1 make a capsule values I still get the ax I mention pull out the same thing the maximum of arrests students a year but there's an extra Mexican bring outside that's cool because remember this is bounded by Lord acts on a divided through Biologics of school in the western less than and now I've got inside the since grew from minus 200 feet of a primer over a in absolute value falls 1 plus 1 of the log exports over 2001 1 .period so however I wonder how I lost it's been hard to tell able complicated at the very least home so here I what can I do well and for instance bounding this psychic cashier I think township because they can hard but a prime mover race squared it's easy to play with 3rd such isn't painful thing that but anyway so I wanted to get up the balance would be to expanded in way and perhaps I don't wanna get into that at this stage but suffice to say that proceeding like this you can for instance in this problem and other interest problems reduces bound to root love extra-large so there is a way I trust me so far this is something we're going to get into more when we get into hoppers new proof of police's their how you can win by doing such things but this actually well under 170 and time today this is going to come in now so many actually finish off by something fun so what we're going to use are going revisit the proof of prime number 13 we're going to do it to the right of 1 would you do it by taking absolute values buildable clever about how we construct the integral but when you just slightly different function from the standard function so but if you can win if you've been lost up to now this is kind of fun to come so I said This is
a farce looks like 1 arrest minus 1 plus gamma firstly go goes best minus 1 when your little 1 and it's it's going to take a series from then on so this would give us was a surprise .period just the differentiate right minus 1 arrest minus 1 plus Wisconsin squared plus Figure 1 thing and then say what interrogators minus 8 to Prime Minister over his 8 s counters that start that starts with an 1 arrest minus 1 plus scam the scammer yet no -minus it's that's just a calculation based on what's now what we want to do in the process of prime numbers Burma's integrate Zeta prime movers agent and the whole issue becomes the main town becomes the Politis equals 1 but in that problem we work and we want to actually get up abounds on the into granted right of 1 so somehow we wanna get rid of that poll but as equals 1 so it is a nice way to do it let's just look at Zeta prime mask of his 8th St classic tests so that's that minus that the polls gone if I'd subtracted gamma this is as equals 1 it even vanish think so I would play this function let's see what's underneath about this function now has no poll Annex equals 1 right we just cancel out but let's have a look at the zeros so when actually wants is 1 of the polls now we are we know that the polls of a Will the poll that this must be at least a poll of 1 of the top no polls a gamma they pose a addresses address equals 1 that canceled with the polls later primaries so the icicles 1 is Interpol my recent contributor Paul 1 of the remaining polls we set many pulses a prime later the polls as equals 1 and heroes is a dress but they remain poles here because nothing during the summit to the polls of this equal zeros was a factor so 1 way to phrase real hypothesis instead of looking for a 0 soaring zeta function we look for polls of this guy and this is the guy we're going to put into prawns formula tomorrow but today I just wanna play finish off by playing with us so if you did analysts you might have good ways of tracking zeros but if you let me the only thing you know that's easier to do with this show something's not a appalled because some example means a dozen by Finnerty which is relatively relatively easy to do so we have function in the call this function ZS and in Bremen hypothesis thing me rephrase instead of S has no polls where the real progress Christopher half right and 101 think about a technique that would allow me to prove that and stuff and it's not something OK when offer goes to harm so how can we proved that a function has no poles in a certain bolster will 1 way to do that is at a certain point is if you have a Taylor expansion with a certain radius of convergence and I think emerges that doesn't have a Paul so how about this for an idea he is a half-life and I want said events have no polls right about half but anyway the players said invested writer once so here is that a mess and I wondered what understand builder Taylor series presented at every point to the right 1 and I want to prove that the just doesn't have a polar what if I can just step radius of convergence are hot for every point here I can include every point of arrival obvious everyone to write off figures Hoffman's epsilon just enough to skip over the line and then hopefully it'll be within the ball of convergence that so the rebound hypothesis can be said St. already have also be employed by Z S 2 0 as the U.S. has Taylor series With radius convergence at least a half and for role as were thrilled by the spirit of anyone so now suddenly covering hypothesis living to the right of 1 so how do you prove a detailed a series has a certain radius convergence well when you write it out you write it out well in terms of technical coefficients August etc. so I want the U.S. mines as heroes to go up to a half so this grows like a half to the I can make this converge affairs growth smaller ventricular pressure grows like to to the case then I can get a barely raised virgins less than a half and that'll be enough of it's an open book so his mind during hypotheses said came as 0 it came as overcame factorial were asked is signup poolside cheek for all ethical simplified Sigma greater than 1 that is bounded by something that depends only on at times to so if I can just take these derivatives high derivatives of Zadar PRI members 8 Jose Zeta and will gamble we know how to do hundreds of all of things but if the 1st 2 terms like can high derivatives of imbalance this well then in preparing hypothesis because this is what we call a pretentious Roman hypothesis again and that why pretentious but and then we just say that if you assume the Riemann hypothesis than just by classical techniques room hypothesis implies that although streets list so although we need to assume is that there's some bound with a fixed constant for each point it actually rehab will come back
and give you a uniform bound and every point so How would start next time I guess is we're gonna to make this assumption and we're gonna prove that return X the half of square that that's only playing on the right of 1 now that may not surprise you I mean we have approved ready because this implies opposes the missile room have also supplies that but I want to build a technique that doesn't go into the critical strip will you zeros then what would you do as well OK we were we can't prove this week come preparing unfortunately may we can prove a weaker version of this something else to the care and using that well we can do that and using that we can preserve the results so that's what we'll do next time and once again that which should the 131 20 minutes and then we'll move on to this more general technique using profits but it was the worst look at what is going on and it is the 1 of the things right he's going to lose the vote approved the use of the whole island I have held months islands known as to what happened to him were there in the answer is yes there is a formula for the political result yet it it takes a little bit explanation so and I don't read overeager dropped off in the distance function will excitement at Internet site question I mean the summit different issues to talk about but I'll talk about this 1 of the 10 the title of year in divided over the top yes I admit I knew I had plans to talk about everything remains memoir comes into the proof the prime numbers the leaving it's leftist the rating unless economic policy was 1 thank you today
Resultante
Distributionstheorie
Gewicht <Mathematik>
Analytische Zahlentheorie
Zahlentheorie
Momentenproblem
Wasserdampftafel
Gruppenoperation
Besprechung/Interview
Bilinearform
Zählen
Mathematische Logik
Statistische Hypothese
Stichprobenfehler
Physikalische Theorie
Computeranimation
Ausdruck <Logik>
Primzahlsatz
Logarithmus
Arithmetische Folge
Theorem
Inverser Limes
Vorlesung/Konferenz
Analysis
Leistung <Physik>
Schätzwert
Lineares Funktional
Erweiterung
Primzahl
Klassische Physik
Physikalisches System
Primideal
Objekt <Kategorie>
Arithmetisches Mittel
Konstante
Zahlentheorie
Flächeninhalt
Sortierte Logik
Rechter Winkel
Offene Menge
Physikalische Theorie
Analytische Zahlentheorie
Beweistheorie
Mereologie
Riemannsche Vermutung
Harmonische Analyse
Einfügungsdämpfung
Punkt
Gewichtete Summe
Fortsetzung <Mathematik>
Kartesische Koordinaten
Zählen
Gesetz <Physik>
Inzidenzalgebra
Statistische Hypothese
Eins
Abelsche partielle Summation
Vorlesung/Konferenz
Gerade
Parametersystem
Lineares Funktional
Extremwert
Primzahl
Physikalischer Effekt
Reihe
Störungstheorie
Rechnen
Biprodukt
Konstante
Polstelle
Betrag <Mathematik>
Sortierte Logik
Ganze Zahl
Rechter Winkel
Beweistheorie
Ablöseblase
Ordnung <Mathematik>
Gammafunktion
Explosion <Stochastik>
Standardabweichung
Aggregatzustand
Geschwindigkeit
Subtraktion
Folge <Mathematik>
Quader
Zahlentheorie
Wasserdampftafel
Gruppenoperation
Abgeschlossene Menge
Derivation <Algebra>
Term
Stichprobenfehler
Ausdruck <Logik>
Spannweite <Stochastik>
Primzahlsatz
Ungleichung
Logarithmus
Mittelwert
Spieltheorie
Fächer <Mathematik>
Reelle Zahl
Freie Gruppe
Verband <Mathematik>
Analysis
Schätzwert
Einfach zusammenhängender Raum
Trennungsaxiom
Matching <Graphentheorie>
Zeitbereich
Primideal
Integral
Unendlichkeit
Komplexe Ebene
Quadratzahl
Residuum
Mereologie
Wärmeausdehnung
Funktionentheorie
Surreale Zahl
Resultante
Einfügungsdämpfung
Subtraktion
Gewicht <Mathematik>
Punkt
Ortsoperator
Gruppenkeim
Annulator
Abgeschlossene Menge
Bilinearform
Term
Physikalische Theorie
Richtung
Übergang
Ausdruck <Logik>
Multiplikation
Ungleichung
Unterring
Fächer <Mathematik>
Nichtunterscheidbarkeit
Einheitskreis
Vorlesung/Konferenz
Abstand
Einflussgröße
Leistung <Physik>
Lineares Funktional
Mathematik
Primzahl
Zehn
Dreiecksungleichung
Güte der Anpassung
Relativitätstheorie
Heuristik
Biprodukt
Unendlichkeit
Komplexe Ebene
Betrag <Mathematik>
Menge
Sortierte Logik
Koeffizient
Beweistheorie
Mereologie
Aggregatzustand
Resultante
Subtraktion
Gewichtete Summe
Extrempunkt
Sigma-Algebra
Eins
Ungleichung
Vorzeichen <Mathematik>
Spieltheorie
Inverser Limes
Vorlesung/Konferenz
Verband <Mathematik>
Substitution
Abstand
Auswahlaxiom
Gerade
Leistung <Physik>
Parametersystem
Lineares Funktional
Bruchrechnung
Kreisfläche
Homologie
Primideal
Biprodukt
Integral
Arithmetisches Mittel
Konstante
Quadratzahl
Betrag <Mathematik>
Ganze Zahl
Rechter Winkel
Beweistheorie
Residuum
Ordnung <Mathematik>
Dampf
Punkt
Gewichtete Summe
Prozess <Physik>
Extrempunkt
t-Test
Kartesische Koordinaten
Statistische Hypothese
Gebundener Zustand
Puls <Technik>
Exakter Test
Nichtunterscheidbarkeit
Meter
Vorlesung/Konferenz
Wurzel <Mathematik>
Figurierte Zahl
Gerade
Lineares Funktional
Parametersystem
Multifunktion
Primzahl
Acht
Klassische Physik
Reihe
Rechnen
Ereignishorizont
Teilbarkeit
Polstelle
Druckverlauf
Betrag <Mathematik>
Rechter Winkel
Beweistheorie
Koeffizient
Gammafunktion
Normalspannung
Aggregatzustand
Standardabweichung
Subtraktion
Wasserdampftafel
Gruppenoperation
Derivation <Algebra>
Sigma-Algebra
Mathematische Logik
Term
Ausdruck <Logik>
Logarithmus
Arithmetische Folge
Mittelwert
Reelle Zahl
Inverser Limes
Verband <Mathematik>
Analysis
Radius
Primideal
Integral
Unendlichkeit
Objekt <Kategorie>
Summengleichung
Partielle Integration
Mereologie
Wärmeausdehnung
Riemannsche Vermutung
Resultante
Lineares Funktional
Subtraktion
Abstimmung <Frequenz>
Quadratzahl
Punkt
Sterbeziffer
Primzahl
Rechter Winkel
Beweistheorie
Besprechung/Interview
Vorlesung/Konferenz
Abstand
Ausdruck <Logik>

Metadaten

Formale Metadaten

Titel 1/3 The pretentious approach to analytic number theory
Serientitel Summer school Analytic Number Theory
Anzahl der Teile 36
Autor Granville, Andrew
Lizenz CC-Namensnennung 3.0 Unported:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/16460
Herausgeber Institut des Hautes Études Scientifiques (IHÉS)
Erscheinungsjahr 2014
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Mathematik

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