Prime numbers and Lfunctions
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Title 
Prime numbers and Lfunctions

Title of Series  
Number of Parts 
33

Author 

License 
CC Attribution 3.0 Germany:
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 
Instituto de Ciencias Matemáticas (ICMAT)

Release Date 
2006

Language 
English

Content Metadata
Subject Area  
Abstract 
The classical memoir by Riemann on the zeta function was motivated by questions about the distribution of prime numbers. But there are important problems concerning prime numbers which cannot be addressed along these lines, for example the representation of primes by polynomials. In this talk I will show a panorama of techniques, which modern analytic number theorists use in the study of prime numbers. Among these are sieve methods. I will explain how the primes are captured by adopting new axioms for sieve theory. I shall also discuss recent progress in traditional questions about primes, such as small gaps, and fundamental ones such as equidistribution in arithmetic progressions. However, my primary objective is to indicate the current directions in Prime Number Theory.

Keywords 
prime numbers
Lfunctions

00:00
If it is great honor for me to introduce plenary electoral professor and retrieve finance professor you was born and raised in L. blood Poland he attended the University of Warsaw obtaining a bachelors Parade In 1970 hand A doctor degree 1 year later already as an undergraduate he introduced fundamental ideas new ideas into safe methods there have been over the years recent years other many
00:46
Exciting developments and methods but they work that Henrik did as an undergraduate he ideas that came and form the backbone of the Modern's theory shortly thereafter Professor thorniest turned his attention to the analytic theory of autumn Norfolk forms here as in everything else he touched he pioneered with New ideas and very great technical innovations as well And in particular Pioneered the successful development of the this theory of analytic bottom Orphic forms to classical problems in number theory has as 1 example of there's 1 might mention that well over the point array conjecture seems to have gone the way of the firm remark conjecture that the Raymond hypothesis is still a report Around and ticket He says Bremen paper tells us in its title Bluebird Dionne sell their prints this It is the subject of a prime numbers the most important applications Of the Raymond hypothesis today and generalizations are the distribution of prime numbers in particular it's very important to study the distribution and their medical progression An important theorem Theories Vinogradoff theorem shows That charm Many of the applications Of the Raman hypothesis they can be dispensed with Professor Introduced ideas which show that In many of these applications 1 can go further than 1 could go just even knowing the rim in more recent years Henrik has introduced ideas which have surmounted the aperitif problem in theory which prevented 1 from using methods to produce prime numbers which was after all their goal this Was thought to be of fundamental obstacle up until this Where he has introduced numerous numerous innovations in the Theory of health functions the Syrian applications of which Are ubiquitous Not just an analytic number theory but in all parts of the Number theory as Henry has won numerous honors and awards call prize announced Rusty prizes for example he's a member of the National Academy of Poland And also work The U.S. as prolific and generous supervisor of graduate students is Annual lecture notes They have become a hot black market items which circulates widely and We introduce the subject to people all over the world and many people have learned from his notes which he produces a fresh from a different part of number theory every year or twice a year It is professor finances lectured this is 3rd invited lecture to the ICM 1st in Helsinki and 1978 the 2nd in Berkeley in 1986 already in 1978 and its was at the very top of the field of analytic number theory many things have changed in the 28 years since
04:50
This 1 has not title of professor utilizing its lectures prime numbers and functions and I'm very pleased to introduce I wish What did the breadth of Ill The docket from members of the functions so that we go to the business estate and the change he farm he examined the prime numbers for price and said that out there and not all of them of course but on read the and many crimes demand particles so of I think so can play with them for ever and then before I go to the subject that interest me and John mentioned above distribution of funds after they won't let me say you things about what the prime numbers so good for them Does that in case you think As the game but I don't think so fundamental for structures It take taken more job in mathematics and for example that lets just take it that they could equation electoral Nolan that solutions test
06:42
These tests is just look at the congress trying for each and If you are lucky contradicts in there so you can go father of cost cuts it'll be numbers and then get back in there solution the numbers the new test father
07:04
I think Remember the completion the rest of it In case And now limited to you the consolidation and do the same completion goes chaotic feuds and they're still grazing jet fuel and a lot of luck trying for the 4 p.m Members master of is that a solution on the roads in the city that is to say are some numbers End of their often dancers The cover of a principle that sometimes so it cost due for other forms of some and so I'll as enough to work the views of the times have basic it in 0 4 4 4 in the of vessel for study on the right there is also the extension of Russian not to acquire knowledge of crimes It these time from ideas replaced From numbers and even those who don't understand you have to understand the best of times it He knows well and that's that does Stewart above it however that depend on this good for and then we often alike can numbers a center Last year And it is subject to play with and I just Gupta questions like What is the largest known kind of gasses that the crystal means largest known when we note that the minute signing up to you to decide what is Mutual to keep somebody trend number But what kind of confused this is number so he is littered were recorded A few days ago and mobile and see it that Largest from number of Overnight but it it is possible the trimmed digits and and the problem all of my choice in this goddess to kind conjecture said He he passed to assess what kind of prime numbers it looks very innocent question said Alone It Often No importance but Did it currently in development of mathematics when we started something film and from just place True with a big game Development father and become basically let things like like like remember of that My last year them The 2 good so much as gage the Kumar Cost decay warrants That are so you can 2 contradictory which is nothing but But the interesting subject on its own actually does have some Then advanced to severe problems recently Another prime conjecture for culture times Can't is to appear in Dublin And said it made comment to the bow to the told the hawkers municipal on Saturday when a tribute to the he is subject sold arms so knowledge The a team of all of the talk about distribution of time is dying men of numbers Actually the letter of intent says the asymptotic personable concept to which 6 objects it's not a positive density by yourself That is 1 of the top 10 future A classes from the from that and that's not much of a Dear toe to toe where proved the density is not More soulful a Carlo and how a that cleaned it was a big issue and Probably hear more about how to overcome this problem in a lecture by by so the there's of Times is While the dollar and so what Lumley Monday that didn't think it you can't send establishes the cut their numbers vary Easily because there are still men of them but the irregularity of distribution is something that we don't call Competent today And still used by being she's still tends set the kind lots of a bailout of problems and did Predictions the prime numbers are part of it is easy to make The rigorous Copy of it so for example it's easy to predict how much will number of Time treaties to the PNP best told us somewhat independent in terms of health At the time of the density of should be over 1 of the X squared and indeed The conjecture behind each with cutting the number of times we see over squares Constant Before I can't because it is must say that it is that it can't find some way is got number of times in just because easier so it's a and out of time and 0 otherwise so that their number to them he is equivalent to today's statement and and was approved by the mother of More than 100 years ago ended the same way Prime conjecture We equivalent on Well we can conjecture is of course a practice on love that it evolution eyes foundation of mathematics it but But the question would be far calm and that dancers Because it is up to you Too much of a factor of the adjective factual Indeed Ended a Is Understanding the structure of teachers is easier these days by means of Carmen cannot Can apply Example plus the estimation formula is he is 1 example of this commission is always on the side Over effect the man The damaged he Integer vectors on the other side of the goddess that those of mentioned of this function press if we France form and a modem modular
14:45
But the formula are due to relations between some of the letters on 1 side And some of allow this side it and the front Is changed by Please ask for a more genetic damage their images on as The test onto the they think this is tasteful last fall on a total ended in marginal situations a comedy basis the parks is committed to a peaceful nominal different today that it has No longer Group elements of side Correspond to 1 of the group of this site But the Chris . 2 on volley of some of the giant operate toll is and there is and it because they would play in the numbers With permission for ministers At 1st We used a bullish entire relations to understand their respect on both sides of the assets of toward To cope must understand a suspect of At this The phone on the business side and didn't play it and you understand completely removed it just for and applications for 0 so far Prime numbers resisted the is the kind of man And do it and they do others object can numbers companions numbers are problems of the functions there he is a function defined as vicious city is a result of drug that was introduced by about owners of quality months at the front end and end their entities It has still total representations very useful for for prime number here 1 Product of the times the stressed that Representation of the complex's zeros and competent integration argument because you can transform these relations into a socalled of the explicit formula on beaches here some over All the essential track and If function is a sample over Over zeros were efforts kind of which are more and Similar many And they're so fine this is it Formula which would rely their protest as a taste for not if only didn't it could be affected by the of some of the signs of But this is a dream which is not yet realized by the very intelligent speculations particularly from Mandalay they end to end A fee If If Well what 1 could diplomat says On the line for here Almost immediately said about But here we may wait a long time before we see these operators and such signs of support cycle that my talk I will be able to show you how much has been accomplished by Grant about talks about numbers and even the go beyond their limits Limitations of the On 1 of the board of investigations in in this area in the 70 s and 80 were And preacher Essentially telling you that almost 4 deals are needed to cut signed we also know wonderful is out of the earned Congress that 40 per cent of On the cover line jazz as for your information to understand some words that quickly sold the quantity of talking about the notion that could cover it the total number of zeros Rectangles or fighter He is a over top I look team for the Atlanta 0 so even more than after he or worse yet so what did they it is estimated that The city has been accomplished unconditionally says that it the number of doses of the critical and partly from the prime minister's very rapidly And there is This is 1 of the treasures of the numbers because it from best replacement of my hypothesis instead us very well for application of the additional of trampling of Over garb speaking time I'm talking about anemometer so that the goal of this project this is in accord with the truth You are not enough time to speak about the intended to be beautiful developments there New ideas But do all But over the years detectors to large by of the ship among us that it keep get At the time so salute because it gives you go so there about our a meter as It is likely to play an overlooked and still have an idea how many of them out there and so you could deduce politically with the other about that Your sister piled up and the Member detect Moment off on her if she said there's pool that begged understand that did did that day They're McConnell All of the beach Nor was The distribution of prime numbers we should progress that is something that is more local He kind of it Prime numbers and and can number A simple because and and by numbers them so the other got concurrence plans but an amount hypothesis would be fuel is the amount enter onehalf look and square between It is far from the average value and The end of the story but might have been 1 exponent is because his you of the front door Exist on the typical I can't get along with But for what it is a beautiful progress that that done about Bob A combination of the metaphor he and the world record here Unconditional love Baker prominent she's a . 5 2 5 closed half it back and then I intended It is to talk about this more but but I don't think I have Right A candlelit By the delimitation of Of the headquarters the exponent and have his best possible because of the color line when Mr. expect better that better results is a disaster that go beyond what comment had a model provides the mother of prime numbers and based on this he conjecture that the adapts
22:38
Johnson look and Square much much better than what offices Can give however this permissive mode both of Over time is Some A thrill It is it is not If it does not have enough The input from Think End In fact he mother predicts but it sounds about At that time It's talking about us each later Show the for to but that might For most of us and so that when it comes to modeling Mr. mother of the business Times that a comment from the From the methods it through use Much more Contains A prediction is amazing to let them at its still on that I think Nevertheless is there's not a medic That is formulas I want to be correct And so it isn't perfect how different worth Of of Are numbers and aunt teachers could drift in here and Explained still on banned Connolly a noted beautiful article and had a comment that on but I still don't know so so so the Dutch The plan should be asked last told that Between prime conjecture that I mentioned before and it is Part of the reason is that it will be impossible him to come closer and his spending is out for 1 year ago The cost of teams who proved that the GOP's Overlooked and The average voter is And actually later proved to be that by by Peter D. and but Founded by look and Essentially infant often their numbers that of because they run The random variables and see for older persons distribution that there's enough life a sense that promise All the way Planted admitted the integration 5 really blocks for constructs instead Tapie England All right but 1st of all I really lament that Basic objects to create thinks the times will find dance another sequence And so we know agreed distributed by a Jewish man who proved that he is the characters and an associate of functions at end backed by means of these people told extended series that functions The men faster automatic it to the other functions including my memoirs on the functions and the airport Set that so that kind of progress now satisfied every class But I'm models is supposed to represent the same proportion of firms and their attorneys pretty good from the hypothesis of course only entities is a need for formula and the model is not to laugh and the model is that group effects which is based in quite quite good uniformity but this time limitation he again that because of the on the critical air we expected mold The and garment conjecture that thought because this would be true for queue opera That actually at 1 month sets And then And I would talk about this later In and and different context alone 1 should not 2 months something distribute there's some limitations distribution and there is a very surprising is out of the could feel the accused us that as it's a little bit the of that 6 for example they ended the asymptotic face on it now have booked his ears Their arms A since the beyond the reach of RO present acknowledging what would The By On the attack us good practice said There's the doctors out of this nature Came from 1965 from here in Grabill who proved that did their best for full 4 progression is still but on amount page but A team of respect for the models after After the size of coupled with X the fact that the same range as they have all this is over A few days the bounty Tests if he is the only state outflow tract and half of what the point being that Uniformity a kind of leftist says that uniformity in in the format it was send to pick our key issue because they are says that dot important complements a that connects different sets of track the way that we would produce pants Fine Council other sequences that around 24 Motyka extremely important and not have This interesting feature that could be used to meet in the prompt is
29:24
A new form of that war crimes or environmental thinker but that functions in the autumn of the function of the demand that this hazardous each of which is very nice comment not note noticed that preservation of his vision of the the investor applications because of this For future nobody has got by the political purposes and the less it is Except ordinarily deformity deterrent to attack the problem because Members of the Segal file logic in this case The village In terms of So this is my mom My mother true With these days is to have the competent something between a abide by it and then I would like to say that reform is not just a sporting event that you want to bet that the league is extremely important is for full for we know that it is ready for some of the things that may look the same way but if you because you know all the Rangers They talk about that But sometimes it seemed that Assad bidding for is completely wanted a different state and NOAA I'm a glimmer of hope is the last word in the distribution of trials is a conjecture that will give out You can call it much farther than my hypothesis could value and that premised on conjecture says that you can get the same statement to us laughter 6 of the and That's what I mentioned because this is credible There will be a consequence of of this conjecture got some teams and ensure that if that could is really show That and that larger than 1 shot in the back consecutive triumph our options abound Off so there's a great beaches In extending will be arrested and put it actually before they did a show that would take a beautiful the lawyer of the retinal some Of what that it is 1 that got goes beyond that is to say we got the bomb early models up to 2 4 7 4 7 doesn't have to see it as a an and ends but that is what we want is major problem here to to use it for what from children is because we do not count their of attention and if such a violent incident which which we too close with durable last time find it here but the issue that these waves up to the day after the time were as good as this thing has really worry about having a value but know is that this is not a siege ended That transparency and a bit of explanation of how this is out of pain because combined so that That meets again and I'm not sure if It said it will explain it time till it means that the wave of transparency will you There some for it but I think I'm aware of our and that there is a lot of interesting ingredients over there may maybe just a few things that The way we prime numbers is much On Romania's 1st if you Tend took proof is something beyond my have what you that bite by using explicit form of reasons you Functions you would like to know that the distribution of on the line and that there is a of Under the front The preparation for example which if it is found that a summit is a step forward to view of the above his but very little at an end and then see amid heavy I would allow a such a statement of a knowledge about does not really reach far you really have to know the shortest what right without there And what they represent So so today they tackle Johnson provisions filled beyond X here for example is is it Users None of the prime numbers and what There is a comment on the composition of Some of the kind that there's patient he met And And you may see River a profile very very elementary years since the beginning May the plan however my point is that what with the composition See you different structures each complement and should be able to apply different kind of crime and others for the fact so it is probably too simplistic to think that you find 1 of the joint operate or the standard time may be coming You go off Operators will be better things to do that Have been composed yield Can't took away so she's is a little bit off off taste of that at all Of the point we know that it plans to buy the inclusion principle asked us on suburban form 2 meters 1 has Is large variable which is to move and that was more credible which is valued at and then because large but but it was still a If we and that is valuable so that just didn't you want still still analysis but the other part of the Composition is is limited mn accomplices of time a mediumsized and they're both here and in here You can't be dead but that is not a very good point here is that it would choose Adonis of Rex remember that we've had in mind the reason for all pleasant for all trees that for me is that handles and principal analysis that if the price is believed that you should apply devolution of transformation benefit from when outside
36:06
Is better than religion out what Social Social that the breaking point here is that this of that would be much stopped during better social this is the reason I'm here in this argument that is of the called equation that publicity equation amazing that it is beneficial Apple plans it was wasn't about point and that is being Jewish thousands of X and end the equation becomes valid and that this is the imminent equations that changed by the operator of G8 introduces till the Antoine OK here we are pieces that this is that The question of Both tontine The disease Did the example of Haitians but that that's emotion from very Feb lots of problems you course Because of whom as to what the value of the demands of the data for thoughtful must forms and there was 1 of the values It said A anyway practice the this way My sport practical both sense of the But do but the idea what saying that that that that that that that the composition the composition of the possibility of playing with different from among the different complement There are is tho mothers is still need to fantastic it even in this sort beating that it would be better in the distribution of bad Then there is some irregularity China's if not goal noticed that the tendency of any more The for the on it For the buyers agreed to visible but said Because it and that there is a series of Davao but not for steam towed Remember it explains things and that the kind that is that there is a wonderful addition publication by the standards are met Wednesday his bison would be classes the detailing measure of all The advent of it the various effects of distribution away everything about this is that in a meeting with illuminates how how the can now must cooperate to The commenters Look at farm When testing distribution of crimes is very dangerous to missing the gun suddenly around on you and you can you can you expect anything sensible But I tell you that but they still move from a function that function is is is that it is the season of the inverse of the top and then it her It is It is a practice much more random variable and then distribution of time I said this most words this kind of randomness of maybe that whenever a village at the time said but maybe assumption of Sequence of integers that not biased Still listed as missing in specific let conceived Where we work with such things would know what the court because the efficiency and from 6 to be biased by their nature of perfection of based The fact that there is safer Safe to assume that maybe 10 signed and and that that would be very useful for testing samples of trying a bit the point is that among those functions could be used massacre combination of maybe games Where something will and because of randomness of the assumptions tourist volume is expected on this more devices And refused to believe that you can right Lots of very interesting close to come Most of the with some of the Giants celestial mean that concedes that it exhibited by and event Tho this quantity should be proportionate some of the and the and the density of function and have of the set and And using the founder of my assumption is that you can we can't predict the before they enter constant and at the of the says it just an affidavit formula Is of course never face and the same formula predicts Is it was forced earlier claimed on what basis of simple met privately with it that it is thought to kind Other things but that's what said the principle behind its diplomats and this is banned the of maybe is very very close Conceded that It continued to the most powerful of the There may be a few days Complete them That sounds nothing about my assumption that area even tho it is that if Otis because it in nature It is that there really is unconditional and decisive achievements of along such lines using metal band that formed at the end of and I think that exaggerated meeting with told that explained in months and it works and I don't think I would do it just so you just something something that for so it is but not Time for it but that is what 1 might think I should mention that The new parks somewhat and is former estimation Please New ingredients it's too too early to work on for 4 years and then be able to 2 establishes asked where use on not available by Steve that is a wonderful to have that maybe it trouble principal insisted the parity program the statement that the alone In addition the Is unable to produce from numbers but because we are we are have no of limits to whenever there is a physics is for the limitation a competitive problems will is just that exceed explained this in the most professions and
42:55
Time but just illustrate what this means but that the traditional method It you cannot produce Numbers leader since I don't know but not in that only can numbers but can numbers which had the same tired of of of prime number brand devised for example cannot produce changes having to power Dozens 2000 a number that you don't produce numbers 2006 president said that he or 1 of crap but its use So this is the core the part problems Sufi reaches the fury and there Examples that you cannot make it that act Absent here that did that but the example and it allows to produce time them and others out of this nature time was but the point is that maybe some random is not biased to this feature Actually it is difficult to introduce but the verification in and traffic case is extremely difficult of and still here is a list of results Of St. as such Lines and it would be like That is at full romance says about hands gaucho pants essentially you know what it means you know you can count it is who we want to go as The best and would be prime number and in the same time 1 1 source of number here is not demands that be found injected because if you look at the density of found that that here it is Very close to the 1 that ended to track conjecture Others Results of my staff within days because this is the is apparent that before it And there and another 1 is still Brown around a cheap bestial DQP Stephen Morse by sequence and it's so these his now unconditional unavailable for the but for the but but but Can be obtained by a combination There is from analysis and woods Toe to with Camille Corot aren't here this is some nice Byproduct of investigation of all of this is because the 4 times the speed Belgium's times it plans the eyes of the independent variables they on each other However Dad That's A little low barrels went with this Kinds So once is cautious about sinking the trend is our somewhat independent objects still can't The from the subject so what we will is that to doesn't allow you to build relations of Johnson symbol of between took Currency but Production times those of 101 for represented a stance of squares and associate 1 symbol of what Such a crime It and put it in the best might sign and it we proved that this team is dead at the bottom of this The sign is that might want call it what it frequently entered the desire to see this so it's fun is 1 of those it is comparable minimize the pleasant but this is It's not Because He's not wanted to get that It is the product of However I didn't and have a lot of in explanations have some members said that the amount of cancelations in subsidies the the number of attacks it is a very foundation to believe so that's narcolepsy still provides that it is very very bad so we apply the when testing things in the but U.S. the coffee Bush because transcend the extremely hers independent in the dark he is something to the Mr. recently encountered some price if like that These are the value of the operator of a set of of projects approved lead and duty Manipur forecast as that of functions A says that we needed just group of cancelations of sound on their demands for the argument that a function of these to cancelation of a group of Member of attempts in sound and tumors that he does not see this exponent shows so when you tweets you should view of abetting Jewish group of isolation and West has revealed that some cancelations no question about that but surprisingly enough not for much on that said the body of the report that is some and the vise And it and in and out of the 6 of function is still can't understand is a constant You know it's a value at 1 of the French They put em Have not much time left for for something that is the best way of both Of the craft of controversy because so that spent some time on that their daily lives Exceptional beach It was not approved loan to go today ordered punch It did another manage a decision he added creatures is are good enough for called for more basic applications numbers but there is 1 exception costs of said Product There is an email to put an end to a for a function of aircraft may exist It is closer to 1 It's can't accept that it is The practice of The azido Italy out and sit there and of course if you believe in a manner but is such exceptions exist but 1 of the more basic It is a problem we face Today the number is to eliminate the existence of state it is the the best in with it in the area of the because in this type of of distribution of prime numbers if you look at the formula
50:02
He and that is very close to 1 depending on the value of titles They people on the best minds by the end of the 2 times more times insisted class It did not normally expected All you have no no no no prime essentially a United States citizen ability discarding the distribution of funds and progress What and tensile OOC me with the cube that let let me just days There is something what is that Dixie here and how we can enjoy actually existed of chips Fictional character and for simplicity assume that character is even resort so it is from symbol of associative other fuel imagine refused And there is a deep sheet of both of the stars Arithmetical that's used that The excess number formula explicitly Relations class Just number is Don't is is Idea Clive Of have difficulty exist Field and the measure the degree of fado A position No doubt very important vote in the The toe toe toe no amount of course and that destructive the test of it says but even the order of the group is for food and a mental don't the start about house 9 unions that have time to he said and that will take a different view point for the purpose of the stock so let me just say they want to on this Very tricky property of their sets out Current and Except 0 is But before heading so the that roughly speaking says The except as you know is very close to point want a say in the speech There I is not on of the same function Other functions and not on the but of prompted euros or any other look at it could be you know additional function would be uppermost Arafat could be as function All over the world Often number fewer except that it did At this year head of state farther away from what for the desert top of understand that these articles such properties because if you don't believe in the form that can be explained that such a pact Pretended to be demand function it takes at times violent minus 1 more often Not dead and the reason is that trust number is is that there's more there's very very very few primetime viewers that out of the London she'll discovered this year so that there is a itself demand from Chile is the fact that there's Maybe function if you remember the use of my function the efficiency of the EU we have wonderful it has money to start the eve of the function because factor the nature of the right It is a different matter is that you know you practice when we don't know at a level that is still member of the team of where he social all the way that they plan using exceptional path of course I have to be careful because we don't believe that But we can't Got so that it these days This kind of the produced by means of its certain practice of you lose draw period a yacht Well if you want to have something that is That that is that this is a really effective and then you cannot work would have put but that the probe assumptions about 0 because we accept as you because we don't believe it exists but the a wonderful observations on the Many years ago that that The end off And function is know There is a success should not be here that have been the sense of could petting proper Pulford heading is diminishing would 0 both to the center appalling but is still there And then the chief from the test nation's credit card much Maine edition estimation effective that of such exceptional as but if you approach would to you if you accept zeros closer to the point or even when it is which is how it is a little bit and condition of his I can't believe you can't seem to find exceptional zeal of the line because of her father's so the next question of what happens is that if you accept 0 the center of report that the hadn't prepared to case said Does the fact that it the job after and the Jesuit The point still maintain that their repelling probe and where there is a bit of diminishes and but it looked completely Stewart magical then it's still shows some some part of And he has a chance and it was voted today it's a vote by the gods such examples of The exception on In that sense it said It still exist in function the preservation of Paris in a multiplicity of interest you know is equal to the amount of more than and deed is this Heading the Feb to the Effective bound the class member of the House around but now Let's assume for a moment Part of the villas and those of Attempt to prove The the existence of all over the exceptional character and however Received from its property is so useful actually did that with the exceptional character and its acceptance of a weather cop who would love So let's see what can be done with actually it is set at power forward truly that that it evenly over the their forms said that it had They have time to shoulder the show on view is that it just mentioned thought tool
57:06
It is now says used to prove that instantly mandatory cracked you using the exception and those John the times over this That he treats by their way It took their postSoviet To cast before we move on the 1 place of that batch sound but did these They noted says his ass that temporary somebody not to use the arm but but it does Where to look in the car I want to hold that's something that conditions alone such idea planted and He said that there That didn't want to hoped to watchful for existence of exceptional sufferers school and it proved it did The existence of such a plan that would avoid not have what Very very badly in the state is said the function for Kraft would want to call the CEOs of the alignment just But complex here so it's it's unbelievable so I don't think mounted by the police said the exception exist even tho it is just not the longer go some people would it may be the case a meeting except The fact is 1 of the most important problem number theory and Let's try to do something and conditions we will be forced to what about I need to be said of pragmatic district zeroth look at classes on the on the Which may be more realistic define you know so it is 0 Sunday 9 is fragmented That will be OK if we don't believe it may be just classes Lots of It Accumulation of zeroes in short segments and he's of wandering myself Recently established He says That is a positive The proportion of pairs of zeros of month The seen on that With probe for the proportion vetoes have got more than half of the normal adopt them is no exception So it is somewhat out of that is that there is the a zerosum functions chart That there is no way didn't we used to think that independent of where on the status of the matter as saying showed that But but about There will be a trade fair if I find the did that today It people's fight so she's this example when you may ask where This class They really where There is a man of letters bigcity they that their preparations after series in the context of the functions and it was done assets This topic which tells you that Said that the number of that is not that can be more often so that our assumptions about what it will take on the pieces but listed arm even more precise distributions of gaps between euros is established by the
1:00:48
Paid and the correlation found so and so still even tho it that did did this Idea of Egyptian It is unrealistic it is sometimes a situation that we get deviations 4 4 4 4 positive for that it works and as such I'm Show the messages There dead When's should try to use found itself of A lot of punches problem apparently is lifted on it to 1 I think if we went to bed with small but released today and they so it off