3/3 The pretentious approach to analytic number theory
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Title 
3/3 The pretentious approach to analytic number theory

Title of Series  
Number of Parts 
36

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CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  
Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2014

Language 
English

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Subject Area 
00:00
Geometry
Multiplication sign
Theory
Theory
Local Group
Flow separation
Category of being
Computer animation
Manysorted logic
Lecture/Conference
Logic
Number theory
Analytic number theory
Boundary value problem
Number theory
Analytic number theory
Field (mathematics)
Subtraction
01:22
Classical physics
Point (geometry)
Beta function
Link (knot theory)
Multiplication sign
Range (statistics)
Tube (container)
Process capability index
Solid geometry
Distance
Dreiecksungleichung
Order of magnitude
Prime ideal
Derivation (linguistics)
Inference
Latent heat
Manysorted logic
Lecture/Conference
Wellformed formula
Square number
Queue (abstract data type)
Number theory
Analytic number theory
Subtraction
Field (mathematics)
Physical system
Area
Addition
Multiplication
Spacetime
Product (category theory)
Price index
Incidence algebra
Multilateration
Line (geometry)
Functional (mathematics)
Connected space
Proof theory
Summation
Spring (hydrology)
Object (grammar)
Thermal conductivity
Arithmetic progression
Resultant
12:49
Logical constant
Greatest element
Beta function
Multiplication sign
Range (statistics)
1 (number)
Routing
Mereology
Inequality (mathematics)
Event horizon
Power (physics)
Quadratic equation
Sign (mathematics)
Population density
Lecture/Conference
Wellformed formula
Natural number
Analogy
Square number
Number theory
Diagram
Absolute value
Alpha (investment)
Social class
Focus (optics)
Product (category theory)
Sigmaalgebra
PoissonKlammer
Moment (mathematics)
Infinity
Incidence algebra
Limit (category theory)
Variable (mathematics)
Functional (mathematics)
Connected space
10 (number)
Estimator
Proof theory
Summation
Object (grammar)
Arithmetic progression
Resultant
Surjective function
27:52
Elementary arithmetic
INTEGRAL
Multiplication sign
Mereology
Prime number
Maxima and minima
Sign (mathematics)
Mathematics
Roundness (object)
Manysorted logic
Meeting/Interview
Analogy
Square number
Cuboid
Length
Logarithm
Process (computing)
Basis (linear algebra)
Infinity
Variable (mathematics)
Functional (mathematics)
Proof theory
Arithmetic mean
Right angle
Arithmetic progression
Identical particles
Point (geometry)
Real number
Ultraviolet photoelectron spectroscopy
Routing
Student's ttest
Event horizon
2 (number)
Prime ideal
Frequency
Goodness of fit
Root
Lecture/Conference
Wellformed formula
Term (mathematics)
Shift operator
Prime factor
Surface
Prisoner's dilemma
Element (mathematics)
Prime number theorem
Algebraic structure
Line (geometry)
Set (mathematics)
Limit (category theory)
Logic
Vertex (graph theory)
Film editing
Family
Limit of a function
43:29
Group action
Musical ensemble
Beta function
Multiplication sign
Quadratic residue
Mereology
Prime number
Weight
Dreiecksungleichung
Grothendieck topology
Subset
Maxima and minima
Exclusive or
Binary quadratic form
Analogy
Square number
Number theory
Circle
Length
Physical system
Logarithm
Product (category theory)
Process (computing)
Gradient
Thermal expansion
Functional (mathematics)
Bilinear form
Hand fan
Arithmetic mean
Graph coloring
Right angle
Arithmetic progression
Resultant
Point (geometry)
Divisor
Distance
Event horizon
Rule of inference
Power (physics)
Prime ideal
Quadratic equation
Congruence subgroup
Goodness of fit
Inclusion map
Root
Term (mathematics)
Queue (abstract data type)
Subtraction
Orthogonality
Units of measurement
Alpha (investment)
Condition number
Multiplication
Physical law
Prime number theorem
Variance
Division (mathematics)
Unit circle
Quadratic form
Limit (category theory)
Convolution
Vector potential
Calculation
Summation
Logic
59:43
Point (geometry)
Classical physics
Standard error
Trail
Musical ensemble
State of matter
Confidence interval
Multiplication sign
Image resolution
Direction (geometry)
Student's ttest
Prime number
Theory
Goodness of fit
Manysorted logic
Meeting/Interview
Lecture/Conference
Term (mathematics)
Analogy
Number theory
Absolute value
Field (mathematics)
Units of measurement
Physical system
Area
Standard deviation
Moment (mathematics)
Staff (military)
Functional (mathematics)
Bilinear form
Local Group
Proof theory
Arithmetic mean
Hausdorff dimension
Right angle
Coefficient
Arithmetic progression
Resultant
Nearring
00:01
In Hey if In you can do In all and
00:13
OK thank you sorry that I might go time try not to but relieved Tsang if you need to leave leave even after you to leave your homes so I thought I'd start by advertising have a special year number theory at the institute this year and some parishioners property Montreal there are 4 conferences in analytic number theory in the autumn semester those the dates as the titles of the 3rd 1 is on the boundary of analytic number
00:48
theory Maris record geometry of beautiful nowadays about the work of Bogor when several strong people like Venkatesh number years that there's a lot of difference between for them harassment algebraic methods analytic methods will come together so this Swiss celebration last November and perhaps more obvious topics and that of the other 3 conference is the 1st time is going to focus logic stemmed from counting .period so a final fields where the size of some groups of sort of looking for a little easier
01:29
knows that will find out soon landfills so I'll get you look on the Web site of here and you can get information information how to apply if new funding also there may be some depending where you from the rest of my prejudices that city with whom supplies money to assist the home OK so let me get back to my topic I want to prove police's from today Linux there today or what strong indication links the I'm writing you some things we've done is a definition of distance is the connection distancing the size of their face serious on throughout the day after small function we have bronze formula on we have Hollis system which is what ringing to prove on a solid start today by 0 talking to me asked me about Seagal's theorem which other exactly want up the Siegel zeros from it's pretty connected on and so what is the seal 0 if you have a quadratic character then separately written this way after his disappearance data that you have some 0 there trail function where data is close to 1 here ,comma is a real character and so on if you just think about the draft of his 1 minutes 0 very close to 1 and you can do something about bounding the derivatives so they'll function ,comma grow much so it matches equivalent to come being small accuser conductor love you and actually would you can typically do these things is truncated well at least 1 of them sort saying all along as if you look at the product yeah if this is small goes down toward 1 of love you the sums the predominant surprised to be underrated plus minus 1 the Coyotes so here there's a prominence of minus 1 plus 1 of some sort of bias so this is the same thing as saying you can make this precise but Taipei is predominantly minus 1 which of cost New so that's the same thing in all languages saying new pretends to be kind risk over pretentious and that she rather more specific release if familiar from onshore wife is all very poorly defined but the media we little more precise is the distance between our the distance square between a new kind and now I'm actually going to restrict myself so on this is P up to actually changes definition there and say if I put you on an interval Q X and here I mean the primes between Q act so between Q and he said the 101 feature From the Beta was a single 0 it's very close to 1 so woman speeches very small 1 miners space very big to the 1 woman if it is really big look at whatever love to all singles here also defined to be within a distance 1 of a long queue of 1 so there you see this as sort of a nontrivial interval and this is the same thing saying that this is uniformly them so there a direct link
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between Siegel zeros and pretentiousness appropriately written out of so when will the catalyst could write this down every time but there's something that distance over a certain range when quite close together a case of Khyber tends to be new insensitive interval now 1 of the great results of analytic number theory use repulsion resort our repulsion results so that that says that well and used some assuming will know well that and can't have 2 zeros of as a century close together and it's the 1 line or even 2 different later functions functions which are roughly the same thinking that they can't have zeros too close together and now if we were to say have to seal 0 supposed to characters supposed Taiwan and kind to head the heroes come and you want them to the conductors so close together OK so of the roughly the same order of magnitude this is a Bulgarian approximately big 0 1 behind the and as I said this is going to be impossible from analytic methods they use various techniques to look at repulsion so let's have a look using our way of thinking of things this would mean that the distance between you and Clive 1 is small and it would also mean the distance between you and kites is small the both of those are small because Siegel 0 but then will have to do is say USA triangle inequality and that's spring than recalled to the distance think I won and invites I would also be small and then just changing things around that means the character Chi Wan Chai to buy is very close to 1 which essentially means that well if we look at what sort of things this means it means that Chi Wan Chai to buy typically points towards 1 for the primes not range and that's going to mean when you form the sun out of this that this thing gets too big so it's not hard to show that Al functions of 1 a small often logged fewer should stay loyal to 1 to the tubing that the conduct of Chi Wan Chai to and so you get a contradiction and so this is how I we can prove propulsion using pretentious methods which is a lot easier than the repulsion from classical methods and this is this landmark has a name chairs pretentiousness is repulsive OK so on this said this is a strong technique because it allows us to to show that things can't be because to close together when they give you another example suppose you had all you had some but just this once opposed the new Mukai straight to have a single 0 but you didn't happen to know how I was real then if this is less and less than 1 if I just think twice that and this ends up being so laughter then recalled to but friend fellow the 1 chisquared and for the same reasons this means that the all function of Chi square that 1 has to get big there is 1 function against big at 1 and that's the reason that a function so that implies that chisquared must be principled Woods chisquared so of us standard results about 0 Roosevelt functions into 1 can be recovered very easily in this context just by such such manipulations In and we will I hope gets approval an incident today and little less experience says that I for all so there exists a constant arises here Linux it was mentioned that idled workers work in different fields was there exists 6 this l such that if they can cure coprime and then there exists a prime the last L such that he is congruent to amount a services or less and less than so there a small prime smallish crimes in areas threaded progression and the proof in all cases comes to 2 cases so in the proof that a hard case on confiscated easy case Furcal talk about very briefly these your easier case which is when there is a single 0 the so as Terry pointed out morning will admit residents distal later is that when you are asking for things that objects and progressions you can break it down according to additive multiplicative characters analytic number 3 is traditionally with multiplicative characters and so if I'm looking for PCI 6 QA Prime's commentary you break it down as what the queue was stretch them now out some high but the ways to write this down often attracted to come by any side x kind In the 2 the upside it's kind son Cameron I use it for you these boats thank you thank you thank you and what we get is that from the principal character this is well essentially acts add the I'm single 0 character so this is the "quotation mark take care I not what minus X the beaches inference user base and now what I said and I
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raised raises my diagram very meaningful diagram but so if you have 1 0 it forces the other repels all the other ones a Siegel 0 so that means all of the other times a small well as some work to be done to justify 1 outside the base you can forget them so this fall only becomes 1 of the few minus kind of 8 x the nature of data so again Kaiser quadratic character obviously this is the you've is minus1 ballet's focus on kind of a plus I have a plusone so when I equal in approximately I'm kind they've plus 1 this guy becomes 1 of the few next minus X would be over so the baby Is is the Beecher is very close to 1 the 1 and beaches more or less 1 and so this thing can be written as could stake out next can we get 1 minus X minus 1 minus beta yeah algebra makes the woman's beta it's easy to the minors miners beetle objects as long as axes into large this looks like X 1 feature missing brackets a woman's meterlong exSoviet 1 Axel objects overfeed Q and so actually it's amusing to note that the number of crimes to its mode Q is a constant multiple acts and if you have a single 0 at least in the appropriate range the same preparation Q to each of the 1 of 1 might speed to you actually get a positive proportion of the numbers in progression prize this year this has won him again more this is a list of his history with dead the rest of a lot of pain so that this is moment pictures of moderate size well yeah it is strictly a few if it is possible to do it but you have to work on there's a lot you can bountiful before 1 last pitch from there to here so on this looks a bit like a serving results it looks like take leads to snare federations execute them and then you knock out oppose proportion of them some proportion some constant might take away from the election Tuesday I'm trying to answer that OK duly converted from finish OK so immediately left now so this is a relatively easy case so as John pointed out I was a bit exaggerating how easy it is to dismiss a little tens upon my speed is really quite small by their 1st class and the former from connection put up that 1 pages bound the blow by 1 over ridicule high over ridicule something to come get incredibly small but can get produced lot smaller than 1 the hard case in the traditional proof is when there's no single 0 seeing a lot of zeros interfering with flight from and they're very very delicate estimates need to be proved and limit dedicate a lot of time for this and the other places and where you can find work on on onto density results that allow you to prove limits it's not appropriate to think so OK so 0 1 the media go now and proved police's there more or less in the Munich come back to limits the so what we said last time 1st time talk was we have promised formula and there was some advantage to breaking the affirmation of the institute 2 parts 2 after that long and blogger and then a complementary part as complementary part we noted let the value of some would be relevant the US and 1 minor slogan of luggage so this is the important part and that comes out in this integral that when we do our whatever acts some analysts route acts of FN Morgan analog affects them by Iran's formula we yeah that approximately 1 over to our and the integral minus F pride ask over overloaded experience arrest and then attacked the edge walls to pull out after that From the integral sure I've come here so this is approximately I will pull out and just wouldn't do at Absa values so this is less than less than unfussy gonna pull out the biggest Escambia McConville overlord acts less than less than an aspects the essence of the Greek replaces gets London's role between minus the and plus the all the best prime mover all seem to excite the numbers next to worry about and that is the duty of 1 plus cases was pulling out the effrontery taking absolute value and as we said this loses an eye I made some remarks about how she went from Harry but now I'm going to wash the Saskatchewan what's so so if we can Schwartz is integral we've got 2 parts we've got the integral of the 1 0 4 1 plus 2 squared what's easy constant and we've got more interesting 1 which is the integral all of best prime mover at square Kansas City Square if the nice thing is you can folded out so the International said here is that will assume again that asphalt peak for the Alpha is 0 if he great that books were allowed to do that when constructing Affin that's convenient when we look at half time of a raft because we make it into a or finance much of it has repeatedly offered to actually make it into a fight OK so what we want to evaluate the integral unless Prime mover asked and so on vast sums well let me work with
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essentially the case where efforts on well I wanted so if we look at half price the task then where is this going to be supported if as has on the products as it does because some of the functions it's going to look like something that's only supported on the prime powers so the anyway and is of prime out I guess the easiest way to write FN rent for the SO it's so similar ideas from minus sign make completely analogous so with data that would just be landed on its own one's arms this is only only supported in less than reflects and nonprime powers and although they I want to say is what happens at crimes which is really going to be important thing Islam the FP just work through looks like after the log so to do that you just think of your product you got the 1 plus FPO repeatedly yes it starts have priming the top half on the bottom and then you see the 1st which is OK so this is the thing that we want take the integral of so it's essentially of land time something OK on land leased the lowkey squads is a little more complicated here but I'm not going to get into that so what we're really interested in is something like following integral as for my speech at the summit up to acts of land that Over end for the signal society of times a events and constant Eyermann which is but in this case at least the prime Sigma received but later on and want the flexibility to little more so and let's do this the please have and what we need is a good upper bound on so what I want to do is see heavy enough among them is known as a slick trick to do well here so let's say but think of functions teachers even but so that the hackers Compaq support patients must be wrong if he is right but what I want is for the well wanted to do they worry that these things this extend this from mice infinity to infinity the same some worry about the imminent and we'll get something that recognizes at least painful between minus the and the OK so it's probably going to look something like this talk and so we have a nice inequality and the idea now is just to expand this the and then we have integral 1 and 2 of once in the IT feed here 10 so this thing looks like fairy transform because each the IT log into anyone so what we end up with years the hat also he log and 1 and 2 this is what I want actually just supported a minus 1 so what would that mean that would mean that feinstein Thank you America yet we just change variables for putting overseen incidence of the so if it supported just a minus 1 1 what does that mean about the the hat tea so if the log and 1 2 years of less than 1 well I guess this means that this is less than 100 feet and so we are going to get in 1 minus 2 is less than and threw it at me for that matter about within L so it restricts the range of 1 and 1 and 2 so let me take the particular case forcing 0 and just work this out so Sigma equals 0 we get out this summer and 1 and 2 all under 1 of the things which will be I wanted to do here is 6 1 of the variables to make the other 1 very short interval so it's convenient just use an inequality like a N 1 into an absolute value is less than or equal to what twice or half the year I think the square this was the perfect so we just plug that in then this cemetery we can choose 1 invariable 1st and 2nd and then it's twice that size can be less than less than within 1 we have a a and 1 squared when the and 1 the the some over in 2 of land and to and substituting but now I should put an end to up to expedite been wiser to put an end to I was in the short interval around and 1 so now we're trying to count crimes in a relatively short interval and so on but we can use From Titchmarsh and each of us treatment that's elementary so there's an upper bound here of lessons some constant times and 1 of the so when we play this whole thing in the please disappear
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automatically now comes the restaurant I'm sorry the statement along with some lenders the government and the good thing is that when 1 of the sorry knowing where notes on scraps of paper feel continue with it and not worry too much so OK mean that the map the value values Sigma doesn't greatly affect things emerges from the 1 example where it will work within 3 days later he was the only 1 of them is that this is a very odd the stressing recalls 1 of the women told OK so you heard 1 over and but then this is the range around and once it becomes 1 over and now we get this this way round it's only 1 option and from then OK if the age for instance alike F that about uniformly bounded then this is going to be is going up to access is less and less than 1 I think so some such a technique the going to work for different segments you just have different estimates and if you put the right thing and comes out of the hands Zoe and might point we I guess in this case that's what we wanted sorry that you use to make point of cerebral world so we have as equals 1 says slander and ever and it should be seen recalls 1 we get the log exits of this question we take the square roots here are out abound this bad by a square logics analog TV but if we take heed be apparent while them the key problem is the square root log cancel things are going so we say to ourselves OK if differentiating once intruding once by Parts 1 and a half way there and integrating by parts is going to get us all the way there that's what you might think but good luck on so in fact and sell goods elements recruits a prime number 34 interpret that he did that but that he goes to a century after prime prime but it's a hard road to get from there to the preferred prime numbers it's not so easy after Celtic's formula when Hopper Hopper came to us with a new proof of Hollister Serrano's Elementary and the question was to try and reformulated in a way that I was more analytic and on the key to it went away was that we actually looked to West crimes over ethanol and enough prime prime so eventually was sound came up with the following formula on which were just proved works and the idea is to have 2 U.S. crimes of arrest and it's a rather Q generalization of Italy's but it works it's important so we go we're going to do we're not into new variables well integrated over a favorite line we had a new variable and we're going to add a new variable basis so what are we doing here come quite finish yet what we're doing but where the hell from data being posed a real numbers and their other strangers ask goes up this vertical line ups the mind seizing his S then in the 1st underground F prime shift along a and in seconds round shifted along a bit more while from so we somehow simultaneously going up 3 lines watchers said there would be few generalization let's see why this is useful in disintegrate over all possible let's assume that things nice and convergence and understand computers that a case of 1st 11 integrate over beta so far integrated debated was left with integral over Alpha Company had been Phil Alford and this just handsome interior data when you integrate F primers out was Baden Baden policy just get between students so we have also asked the South surplus infinity so far to the right award asked evaluated at infinity you go off to the right side of playing and left is the 1 Everything else disappears and when they trickle 0 yet effervescent itself so that's rather coming because it multiplies through so I guess we get we're left with after Prime a listless minus f prime over progress when we do the data integral now we do 1 more integral and well again it's the same trick for the I F prime we have prime becomes 3 of restless infantry mines 0 COS of 1 half Infinity which if you like answers around 1 minus were around half of last year and then on this guy the F prime over asses you go out to infinity and all of those kinds of immune 0 things left is what happens at 0 and so we're going to get my plus log of effervescence it's clear from experience Frank and so there minus sign wrong somewhere but let's now just use on you I'm sorry you might like here cocaine now looks prettier becomes ever less minus 1 minus logger for that so the 1st couple times expansionary blogger from so now let's just use bronze formula has went
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infinity and we get some for enough to ax the 1st time gives a settlement the 2nd time gives us a minus 1 worry about that 1 2nd and we'll all get after that gives you land s event overloaded and then in fact if you think about it the 1 just knocks out the 1st half so you get so this guy is not going to be important it's can be small and so you've really got what you want which is after them we an extraneous term but it turns out to be small and what we have in the middle here is what we want is to have friends over at for once we pull out so the idea here is simply to what a given that the couple things 1 of which is we've got these intervals going from 0 to infinity from beta but it's not hard to imagine that what you think about the sale functions as a go further to the right especially if the referee as as you firms to riding in a smaller and smaller not run so for a small price of a smaller attend using copies and cools down to Mingora sure about length that Epsilon and the important thing is that once only since March this year the same maneuvers but you end up with To have crimes Over so I kind up in a very precise and see that radio but what we end up with is something like that so on 1 of things we want to do actually is slightly moved a line of integration because we've put things so far across the managers say will will cut Alpha Beta down too I find at length sets without any real cost and then we're going to do a slight change of wearable which is we're going to move a line of integration as follows it's not a big deal and when we do that and we do things study carefully we end up with I'm 1 to the interval from minus the to all of the strong and up to some .period X on land prosperity and when their family Over em To yes Men straight the safe remove seeds to see myself my patented to the first one the line goes to on Cminus they to and in the 2nd term we did some of it and and then we've got an F term here and while some will want the capsule values Thompson F of X so what traffic problems I guess we'd do want doses for moving the control and to itself In the absence of any thank you figure that we've got access to the sea 0 minus lines feature 2 T the times with a maximum will even be more precise and this maximum over FC is happening in a box right it's no longer just happening online but instead the maximum that happens all some sort of boxes the maximum heights my the and wit but 1 and so how do you find the maximum for such a thing where you can use on Maxim modules principle if you like and those matters going occur either in the left hand side of the right hand side of top righthand side it's far enough away that going to be bounded and operate difficult not very harmful the Sinochem the importance of the maximum and that they fight so starts botanical at but you end up with you you can see the structures emerging that we want we ain't yet these 2 landers there are some technical issues but it's convenient to make sure these primes and get too big so and that didn't get too small so 1 wants to carrot cut in a bed this will be written out carefully In the book and wants a cut in a bid to avoid some issues coming right with extreme To do that .period also is a little bit the identity approved up there have to begins small Prime Factors dealt with in a way that allows 1 to avoid any these difficulties With this when we take the main square using dilemma that we can approved while you need the extra flexibility signal when you work it all out last works so we have this 1 issue about Russia problems as we were about to find only to Florida so this is roughly how proof might go and resisting the main elements and you can see there's a little bit left over with the 1 over 1 because town that gives a log the performance of logo next if 1 gets balanced suitably here from the surface here and I'm amused bounce that come out of this maximum and they're also bouncers to shift more to the right but just bounded by the zeta function in fact then on you can manipulate that a change of long Peter along the maximum value is from in it's a technical thing but we worked out OK so that's roughly the new proof of there and had it takes a little effort to get your head around that Anderson have been a great job but I think when you read it you will let you get the rest of the main idea I don't think it's so hot is just these 2 F primary refs allows you to use the means square you win the route of X twice I'm not going to do this far but there you have a point from exploiting the sex line itself video will give you another log on him and year it gets to the 1 In the face of the of Academy of move on to be the idea approved limits on which you'll see comes very naturally now and then and you want to start rubbing up here because believe that some of them so I now what we said that the last time I guess is that when you're trying to do the British prime number theorem instead of using lavender and for most of the function person you look at some of U.S. friend to and similar narrative progression 21 propranolol firm for Prime's and progressions and you can look at instead respect and then with suitable
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convolutions if you like you can elementary show that if you can prove that these are all little of execute then that's going to for you the prime number 3 of progression With so this is what we're going to work with but I want to do the same thing I did in the case of prime number theorem is I wanna get more generally and work with knowledge from offered to function as inside the circle so we're interested in an open received here just to have thrived Calvary much OK so this is what I want us to make something to Lefors new and then I'm going to decompose it with the characters and so I'm going to get some over the characters mind you all of them around archive and the sun of any kind or another at the end of what you notice is that this thing is mostly the function of the product multiple functions inside the unit circle is a multiplicative function inside the inner circle it's good so we can apply the technology we've created there and it's going to be convenient to apply that technology but not necessarily using every character here so you know what a shame said earlier I meant to say was when we were proving this repulsion principle by showing that that there was still 1 several things to say so analysts said so this just what I wanted to suppose that 1 of these big case post of 1 of his sons is greater than great for them In a minute will apply the technology malicious because 1 of his sons grew angry annexed the is there and that means that an investor must be so tolerant place but collapsed I am EST but yeah L grade of angry about 1 that's anyway from Hollis case that implies from over there that there exists a less than or equal to the such that the distance from past kind not to enter the IT UP TO X is bounded bank which shot just gonna know this is the same thing as the distance From Past to clients and insecurity now that's supposed to have these terms were big if true these terms a big you'd have this small and you'd also have it 4 the but then we can just use the triangle inequality again I will get the distance between what kind 1 kind to and an end to the 82 minus 31 1 to X is also small and this city like tells you something like the value of L 1 close I 82 minus the 1 at a time when kites bars too big and so we get a country can so again this repulsion principle through pretentious is trying to involve the says to the 71 of the sums can be large and I guess this is Terry little thug analogy but it's the difference here is that Newsom optically working 1 modulus and enlarged so ,comma yet so now we know that there can be at most 1 big guy here cancel let's pull the big guy over to the other side so we're going to have the big if there is a big tumble put it over on the side and then we've got the sum of all the evidence on the righthand side so that's so easy is on all of are July molecule minus perhaps someone was that character summit will lose some of their friend client is big if it exists so let's let's try and boundless using the same technique so we want balanced sound 4 1 0 the some over kind III which is almost everything the forever almost exceptional also some analysts Enrico to acts of effervescence think now what we do here is we just proceed as above a bit of just rubbed out but don't let media and see that it's very similar so on this is less and less than a century 1 of the queue some of the kind I forget the Alpha Beta into roles 1 of the just use grew from minus the the also something like from the song planned S plans borrow them over and while we work it out but minus the true too come and planned and and 1 plus feature over 2 OK cities the 2 from arrest we had but now the twisted by clients in each case the but comes in the 1st lady thank you so now we're she but I think maybe you can see what's going to happen when we tell she wouldn't account putting this sum of client side and and so when we tell chic we get less and less than an action with this whole thing and 5 we get 2 Of the form 1 over the for some time this summer and when they found I bought an end to the oneliners stay true to classified the squared tea and then a similar term here should be a big square root half and now when we expand rather like we did up there we have something extra going on we have the solemnly we case and full characters such saying and it'll characters with something else going on we normally Halverson expansion but when expense full characters wouldn't have em in the same as progression yeah that's what's going to come out From the part of some that's some when I spend that kind borrowed them Clyde and 1 of the 2 that part of the expansion will restrict me to and congrats when 1 so the
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calculation I did before when expanding that I ended up wanting and 2 in a short interval around and 1 between and 1 of my friends 1 minus 1 of and 1 plus flight and between em times 1 minus 1 of its feet and and times 1 plus 1 of the T and from Broome Titchmarsh I got an upper bound on the number of such crimes but and well with the land the weights but now I'm adding an extra condition and congruent when not to but you can also apply Brund Titchmarsh with this additional conditions as long as you have the length of the interval is beginning to use the 1 process so when you do that you win an extra factor I think so winning that exdirector of the Q when you go all the way back ends up giving you something following his stature from through efficient gives you something like that the staying and things will stay in the middle of and the discovery of the rule of law and so what we're getting then is that the sun over enough that it is going to be were essentially less and less than hour up to some logs but most log of an effort to save the maximum was quietly but and this is right this for the maximum with the less than refills tee balls I ask twisted by kind wonderful site overloaded OK so that's when we restricted some subset of Marja like they should be also a lot of this thing as well but the point is winning that 1 of the few means at 1st sight you think Wolfe replied Hollis we just have to some the result from Hollis over all of these Margalo but the orthogonal the conditions with the characters means actually have to take the maximum of it's a remarkable win so now let me just go back to the beginning is that we had the some of their fan up to acts we took out the plausible big character that the no 1 may be big if there is 1 thing to say if all of these Best logic so small than we've already got Linux there others an accidents yes all of the AFC I want society if none of them get as big a slugger acts then we've already won I think we've really got a good but on it's possible that 1 of them is they would prove you can have 2 of them the big by repulsion principles so if 1 of them is they would put them on the other side and we get a good air attack so what if we proved we've proved something like the following is back on some analysts and riddled acts of I don't think Oh this is new of course are rescue of an hour after the landed my head and is like 1 over the queue kind of some analysts and Rex Stephanie high bar and fact from winning a return of the former next time something like blog log acts introduce carefully to some power witches 2 2 it as so we haven't quite truth serum so OK right now I've done it for general excellence is nothin progression it's so the real question then is this what we want is a effect was new we want this to be a little X so we want to know that for that there's no character for which the suspect so let's suppose that the PNC VeriFone progressions fails and here notices home is good as long as X is a big enough power of Q 4 x equals Q big power then when we went through this we have something like I'm sure there should be no you we have a sound the analysts and reflects on you when kind or event his the then this guy is going to be great of angry for them so if that guy's remain and greater than X we know that From Klein is new potential soul music hypertension units from Hollis the but at the start of a pointed out that that's equivalent to having so what we've proved his unless you have a 0 0 police's there just using orthogonal the monkey gives you Linux there and were reduced to the Siegel so just to finish off How do you do if a single 0 case what told you the castaway let me tell you another way this says that the quadratic character Monkey but for most prized for most PC up to access the quadratic residue symbol with of appears minus 1 member I said if you have the a single 0 than a pulsar proportional numbers nascent progression tend to be prime appropriate Arafat progression so How can we explain that while there is a nice way in the Book of Friedlander variance in the Bible the opera the career landing inside more 1942 which is this sunny what's in certain cases but it actually motivates what if you look at a binary quadratic form a square has been quiet CY square which has discriminants Q then it's elementary number 3 that he divides that implies security it's 0 there 0 plus 1 so let's see if you look at the valley's this binary code graphic form most of the values well they come most crimes cannot divide this subset of Prime's it can divide that is a spider quadratic form a very sparse without so sparse that when you do this in the most simple method the the inclusion exclusion you can more or less prove the right number of crimes using that you need to use a lot more technology thing in American history as they so that's what is a simple case this exceptional answered a select that concludes approved limits their they using color system would you can see the ideas
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over the main ideas you can get the details of what hopefully be cleaned up in the book b the conference and you want be the only 1 of the means used to remind me of well in Hollis system from Treasa pointed out for use in this specific case we can't get a better error term In very special cases coming so that in some ways well and I were trying to do just right in this way is that this is a technique that works for General Mills put the functions and progressions when you specialize to new then you can expect a lot more and we saw in the proof of the strong form the prime number theory there are techniques to exploit the sort of thing the a better and whether or not those can be combined here I don't know so perhaps the next question which preempt is how good a Linux can yet what's the value value get was for someone here to do units the professor settle down and then 1 can look for a constant but there is some hope perhaps that by combining these 2 methods you could do well on the constant but the current correct reckless 5 .period it was a colonel in the great code it's 5 years from now In the end and get a better result right it's engineered anyway so there's there's engineering work that gets announced 5 points something and that that's an awfully that's an amazingly Strong Resolve and to do that by any technique is extraordinary for France paper what you be blown away by a number of techniques that incredible that paper and so I don't think there's any great hope in the very near future wouldn't be competitive but units something as a clever idea in Philadelphia has been held I wish we had 10 thousand I have read if we had a number you can write down that rejects the idea that we haven't track we had a previous proof that was much was somewhat more difficult With the old version of the proof policy and trying to write a number was struck some enticing so you really don't do that to you ,comma cleanup approved owing to clean up the proof the problem given constant then you don't have to clean proving look for the bits that are really damaging the constant so it is the work for somebody energetic and good technique that would be good for us that is a problem area in the home of persons in in the world this year having that unit what is the case of the disease and world working to through the hole in the window of the moving and so have a lot of things that have to not be the the confidence that we will not let you down the fact is that you don't want to go back much of anything in the world I don't know is in the hall and someone told you is the head of the soldiers the information is a strongwilled woman in the world is used in the new it's still needed and what is knowing it's the 1st time ever achieved enough that so no it's it's a good idea and I would hope that we uncle identified what issues ,comma could pass group ,comma presuming deals would be a big hit his homer it's certainly not rich and others from using them thank you that was the no 1 would assume that there would be a bit of analogous there is his best this year ,comma refusing Moffett functions should work on his I think would we suggesting was from company on a case so this is something we've we've selected some blood over only to come through a painfully simple conclusion so the it turns out that the techniques more or less although through fairly easily not if the function is bounded by Tuesday but if you take land there case of when you take the coefficients less crime and you look at the absolute value that should be dusting capital and Sunday and this is the pretentious dimensions and the whole area of friendship and whatever so when the band like this then it turns out this is the right thing seems to be the right thing to do the proofs naturally want to behave and the beaches and sound developing some of that theory at the moment so 1 thing we did do I recently was the analogies and function fields partly to try and move has simplify some of this on and what several different functional of is replaced by Talisheek and tell she behaves rather nicely for example and the other generalization that spending should be done for obvious reasons is to go to wait to deal too all functions which I hope somebody will get going on in the near future is not very obvious how to proceed but there is some ideas around you I will say I will see to is it that there will be result OK we may not be talking about exactly the same thing but I'd like to hear offers may be 1 way of also the question is on the right I please and that was it was right here this is not the best of it it goes to the head of a 100 per the state of the national standards that isn't as strong as the most it was set up in the the it would be it doesn't have enough of fishermen as she was former student is quite interested in the splitfinger question keeps on threatening to work on it but he has done so that will encourage him the suggestions from wars it is a
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certificate that afternoon what sort of get prohibitively policies here said thinks cancelation of last year in and year out 1 and of about life here using this the other developed while To look the nobody you'll be nice me other things you know I mean 1 thing they're trying to bring out somewhat but not fully in these lectures is that there is analogy of a classical theory and 1 can hope was ideas of food 1 functions of generalizing different directions and a lot of the theory is after all you think of G O 1 and do something suitable elsewhere but hopefully the potential staff can follow suit and allow 1 little more flexibility so amended it's about the reason something might work but we don't know of obstruction of justice didn't exist and it fills the house once the other fumed as