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3/4 Lfunction
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Title  3/4 Lfunction 
Title of Series  Summer school Analytic Number Theory 
Number of Parts  36 
Author 
Soundararajan, Kannan

License 
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  10.5446/17057 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2014 
Language  English 
Content Metadata
Subject Area  Mathematics 
Series
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00:03
Hey if a and so
00:14
foolish begin by finishing this stuff could have the proof that essentially the argument that here or any so that so we want to I should just explained how 1 can evaluate the reason why they would come to give you this right powerful for over so let's look at that the lefthand side for so I can expand any kind of became minus 1 it would be a sample numbers up to so it kind of goes up to the summit and goes to acts so multiplied OK minus 1 of the most humble subjects so that he might want some kind of function which is like a device function with quality minus 1 and or idea and this function lets just say it's a number of ways of writing an on the part of a woman case where the jail most so now if I use this is not formula that we know that we can evaluate the 1st moment possible but so I can compute but the number of some his commitment of the blacks that this behaves like this assassin public to some constants Quicken which will forget some of and going to exit a minus 1 intent and then that's right and less than 1 times and 2 squared maybe where and when the square feet and when I get there From the form that we still have as the answer so we get something like 1 of more than 1 extends the the effects now to get this up about this a long on for the kid moments I only needed more money for this quantity Everything is positive so I can just focus on what what comes out I want and not worry about don't want that I don't have to worry about this is that at least make as big logic so this is not some next time slot and let's just worry about and that's less than than expert so that I don't have to worry about what decay minus 1 of Annex it's just became 1 of them so that these lodges and act the came 1 of them and then it's going to be divided by then and what times entered the spiral of and 1 and 2 so
03:24
this is essentially a multiplicative function so let's say well it has some of the functions of a minister I put the money into squares so you can see that this kind of roughly looks like excellent at some plants and 1 goes up to maybe tracks became minus 1 gold and 1 and won under the terms of the decay minus 1 and 2 squared Over and this is not strictly speaking because and 1 and 2 can have something in
04:02
common but that's a very minor mind and it's a simple matter to evaluate these is the 1st give me something like X blog little extra care minus 1 on the 2nd 1 Canadian worked out what the what hour flock should be decay minus 1 Of Peace Square and that's the important thing to calculate and that's OK minus 1 times for 2 the power will be will be that the suspect wanted him landscape minus 1 or 2 and then Alexis a small power of capital for the excellent excellent work on the case and so all of this before forget the dependency on cases largest excellent x this is a lot explicate the museum and what which is which is what we what we wanted and we have to do the same calculation for for the for the 2nd time and again this is not so bad some of the fundamental concerns of blacks and some and to explicate the decay can act the rule and once again fighting to change the sum be and some on the square values of animals survive and everything else will disappear so this will be bounded by In the end we just use the function as most as large as the kid was a function and once again this is bounded by excellent text so that's what we will saying is that the the function and have tried it doesn't really look like this approximation HIT adjustment at which the British occupation of of the defining what as as the size of the objects are concerned their moralists have the same size and so would the 1 when I'm using this inequality this funeral also using In this to all no 1 can make some refinements to argument so I said you can also prove the same kind of fruit a rational number 1 so that they think you can write our war I think 1 so this is the truth so the trick in that Wallace was set up on the inequality the way says that you only have to evaluate integer powers of football evaluating a fractured porous hard but it wouldn't be a priority expanded out and then interchange comes into play around with them so we want to do something similar on the idea that is so I don't want to not take the case of Heidi to the king because that's the fractional tolerance facilities gentlemen handle but it would be nice if any of this itself with some power of some of the benefits of the mail right that if that were then that traces the K would just be be of some kind to the some our are so honored and that would be something to the power of difficult and I can handle everything and everything exactly like Lexus said now the scope easy to understand what they were they were just the coefficient 1 tidy and over time so I need to figure out what would be something that wouldn't here which when I take the power kind of looks like you like 1 and that's easy I should just choose the cooperation of the festival 1 of lessons about the case was 8 of the 2 1 0 and that would be the 1 were at the end over was the 1 always adviser functions so that's all you have to do so you define of Heidi to be the the 1 over and over again and again and goes up to some of us were a small part of the poorest cast light on them expands and play around with With the somewhere else To the 2 came minus 1 still into the Power of 104 formula and responded by some more 2 and this is something just for the ship and everything goes so now you can look at that and see but the 1 thing that I know that I have to worry about him is that the whole is how much next week well this whole thing should be a small difficult and the and the size of it's B of title to the power Cape Times which is the famous so I really want that this little acts should be taken to be a small Paul of Catholics but something like 1 in 100 just 1 over 100 right shows many consider this a small problem because you have the number of cases where close 1 but if you your case happens to be hung model 100 it actually is you need to choose a much smaller than facts than you could for equals 1 of course so in other words when you can build this argument you get along on the right order of magnitude but the long bond depend upon came under the upon the height of care rather than just on what the value of so described above briefly but the result you to maximum reservoirs which fixes the and gives you about which is continuous Inc so already for the this was new and had misstated just for that but I would get something like completely explicitly break for lunch to something like exportminded home some departments on came to the right :colon case square case should be left you the mind case 4 so this is for all we the proof of I kind of like the proof because like my stroke I you can use this thing called the simplest of sequence to write what has become like 1 and onehalf of 1 . 1 7 on whether the stage you multiply all the previous fractions on that 1 so rapidly increasing sequence that sums up to 1 the I'm going to call 1 of the 1 you have to use this but this kind of fun analysts say the same thing fall 1 minus 1 of K I'm going to write down some sequence of natural number whose reciprocal to add to that and I can do this and greedy where such was the case with the
13:46
largest number you can do in so this this sequence of numbers the edge of the Bj the increase was rapidly like the guy who was subsequently multiplying all the previous number many 1 but increased suddenly thought much faster than so now the idea is going to be there that we're going to some versions of these some of these functions b of Title VII of the for the zeta function taking shorter and shorter things but fall but police stopped parameters Biagi the beach at home explained and start with the degree of data have cited in the world L goes from 1 to 4 so cold case of Bell of the world I want to peace of mind the and days of Belfast is going to be the some of the of cable work Of over and the S and goes up to some small power the the tee the 1 was on it and then this be so well have minor it's going to be some of the OK over the years and from the again this and also some very small tomorrow for the ideas that is that this kind of stands for don't westerly can and this is the standard for was here for the staff have mine likely to the the sorry so when I'm by all of the powers of the House this guy is a for Zayed office product as standard for the half minus IT to the case and this whole project it's like an approximation of the laws have "quotation mark came minus 1 so so if that is that and you should describe the water written down is kind of like a proxy for the 1st of the 2 but on the other hand the but on the other hand the girl shot dead 4 so let me explain to because the Chileans Guler separately to the wall 180 or so when you multiply all out still goes up only a predictive of 100 the same up to him he sought to shore up the 4 corners of the multiplied and then you can use holders inequalities have to say that this is prompted by the 2 the 2 of them I think you you just can't the part of the world of Montvale likely to lead 2 you can see if you don't believe that the quality of the strength of the exponents the 1 so that someone over to the sum of 1 or 2 aerial adds up to 1 minus 1 over cases those 2 things have to have some of the 1 or 2 appeals had gone to inequalities correct then you can you can again evaluate everything in sight because their function multiplied by 2 short football mills and this the shortage Maryland so everything can be evaluated and it gives you lot work to get more moment of which claims depends on the assumption that only depends on the ground that and doesn't depend want international approximations so let me just maybe roughly explained why that article result and I want to say this In March the day but the point is that showcases the feud when you think about this summer and to the detail as to what these people when you expand things like those of you who should know like I think we are making contribution was coming from the decay and square or an and going up to some up to some quite like gossip the that the constant binds the looked into the case square the fact that you from getting at something like 2 the 1 over 100 there might roughly that when you can yield some expansion like you might know something and have only those terms with them going up to some small power but what's the loss in doing this would be a symbolic something like a loyalty to the case square with the usual constantly but then divided by wealth is going up to 2 years undergoing a particular 1 100 Dale something like 100 failed so you you some constant which is going to be some power or some takes power on the air so we will suddenly lost something when doing this from but on the other hand I'm not using that the power 1 and using it only the part 1 or 2 so I I have some but the losses really going to be only the number 2 1 away to where I want most is not really a L looked like a to the
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case spread a he the life but that's OK because sales rapidly increasing so when want the more or less I only use the constant when I take the part of still the I don't use by more than a constant factor so when you when you expand the size of the upper bounds of began fall for this quantity His only off by a constant from the Obama for the consequent over an understanding of the law amounts for the news media costs that that means that you can log on to formal so this is an idea which is that in some ways related to the city it's also going to be an idea that explains it what ,comma on the upper bonds which I talk about next and that you should really think about this so maybe I'll try to make this knowledge of the sophomore more clear as well we should think of it as if you have to say some number of crimes if the initial plans that make the biggest contribution so you won't be able to support the initial prompt really well on the which of for the people that you don't have to work so hard to serve them because they make progressively smaller contributions so the small bodies of when we kind of get you know but the big Watley the of care which will fan or an earlier we get them out of the the way we take a long long summer but for all of the ones we have to shave you don't really they contribute progressively less so we can the argument that in short the situation calls for more bountiful functions is pretty well understood except for 1 final point which is that these arguments tell you that if I can find something like the 1st moment when I get more for all the larger moments to do 1 thing that's remaining years 1 happens all force formal moments so for example the argument that I gave you for but let's say let's look at Milford reflect that you can look at and have the right to care overall climate cute primakov then the argument that I gave you would show that the case at 1 so if you start by saying something about the 2nd moment here and there to say that it pays at least 1 when we have the right right now but what happens in case of less than 1 so in general we actually don't have good law wonderful moments in the mall and the small range but this is 1 example which we do have a good bonds which were worked out by the Champions the right so they can develop I think when case rational number but you can also incorporate what Max and I and probably get my result for all real values between 0 and 1 so this seems like kind of a technical so I should say that it actually is I think an important questions so that we give you an an example of something that would like to be able to prove results like that but we don't know but fix a modular format and look at the family of the adequacy of like kind and it would be very nice to give law for the so we know there's more for larger than 1 and Bavarian eyes to give the same announcement came less than 1 and the reason I think is supposed to have been the right unit for small values of k then you can imagine living going to give voters a if you like good deserve that function is just picking up essentially what if the values of the world and is picking up some of so if you have the right law Monday as scapegoats as you would know that proportion of these values so this would if this is true for small values of the this conjecture would imply implies also reported on Wednesday to it's it's expected that this value the same value can be is essentially only when the sign of the functional equivalent are so that happens 50 per cent of the time and for people of the times and promptly questions Parts 1 and we expect that most of the time in that case is nonzero but we don't even know that apostle proportional lots of but that will be used to the problem of nonmonetary the problem of these small moments are quite close to but the way you prove nominations do something called the modified about and be able to compute the 1st and the 2nd moment and and then if you can add something called a modified they knew it would prove more mention results we can do that in the case of all all deflected and this is what underlies the work of the elite is essentially they're using a kind of water a modified to get lower bonds calls for small moments which we don't have in this city but what I from this small moments story for a long long pretty well on the case and asked me to talk about the the problems of the moment and also connections to 2 to the center room so that already mentioned for many to think about that meeting I tell you 1 more way to think about
29:58
why we get the various powerful law in the distance in this moment conviction that to that we supposed to do so let me start with that With the recalling some books that I mentioned yesterday which says that if I so it recalled that it says that if I look at the longer they don't have a slightly thickened the real images of the look at the real part here and I supposed to varies between twice captained the team and quite quite that in this range this is approximately With means and what this means is that OK so that's that .period down so this means that if I fix any number among them as ghost and the measure all of majesty for which launches its have bigger than the Times was 1 of the area but tends to the OK so this the appearance assets as it stands is appearing for fixed values are of and as the ghost of when you can you can make it fall in incident but due to gain some insight let's assume that this this kind of behavior persists for all values of for all this was follows the case with the bigger you 1 reason why it falls so Matsumoto something about maybe "quotation mark the large values and small values so you can imagine that the being very very small like that maybe going to minus infinity and in this case you shouldn't expect too much uniformity in with the like this country can hold because suppose you want to that this that you're making some claim about the frequency with which worries more about something like this 1 where would you can produce more voters of data you take is the only take small neighborhood under their own you'll get worse over the few take that you will produce more small values than would be predicted by some of so you should not so you should not use so within that range not true loss small news but maybe you can assume that that's a supports it's true for enlargement of data or maybe it's true an upper bound so that you can get a balanced on 1 side and normalizes slightly differently so let's say that I suppose we assume that the measure of the 2 With Zeta have society bigger than some number at the that behaves the roughly like responded about something like exposed minus the square but am being a little crude here this is exactly the same as thing bill that I wrote down except for the flight normalization I made it clear to the regional leaders of the times as part of the variance on you know this interview roughly like he might be squared with 2 not exactly but roughly the same as it was good and that's what I have written down here again if you take into account the suppose that something like this sir so if you know that you can immediately get enough money for more so this will imply about from moment quite easily because what that took a moment of of 1 the values on which this is larger than somebody that we find measure of the facts on which takes values that are as so this will be the integral from mines and the utility of the times the change in the measure of such large which you can integrate but parts to To be able to use the power of the essentially the same as all maybe Rabin with a minor but because this is decreasing function so well that you did it apart to get up to minus signs which cancel so amount suppose you have a uniform upper bound for the measure on which Was can be hard you would get bounded by and you can just focus on what what is the range of values of wheat on which this is as large as possible that's a simple calculus exercise and you can see that this is maximized when you when to be case and for this value for all we know
38:03
that this integral is itself outside the law the where so you know it's not want that to get moment of the speaking of this to took a moment is really picking up the values of the offside locked into the into on the set of values of data on size of the team to the conducted became the center has measure which is about the only locked into the case so therefore if you take that to the moment of of this you get locked into the 2 cases where times the measure which is probably 1 or 2 which is like so you can see that there is this kind of an intimate connection between still looks the on on the value distribution of the aid of have slightly and the moment so you but the economics of the Keating state conducted what would be the analog of some conjecture on the value distribution of all the love and support functions which would produce producers and exporters of farm gate landscape of to work landscape get and these are just due to the marketing inside convictions of 2 to take that support would take although El kind and the variable fundamentals components state the 2 sides and we want to know the distribution of you can see that this is already a a more subtle problem then somebody for this because was OK I at blogs they don't have Poseidon it could be that they don't have society's What is the only authentic measures measures so I can ignore that and any statistical calculation that I but if you have a problem because if I look at a local have end have pretty happens to be the world's very often then I don't get a distribution of I can't ignore these these values but we knew conjecture that there are very few values so far of all the which I have ideas is the if I pretended to that their mobile of the for which any failed function vanish this is never expected to be 0 but if it is a role for a long time and we have a problem OK so to you can't we can't prove that this number 0 or 0 per set of density 1 so that's 1 reason why these Hamilton opulence but just conjecture but look at the now quite nice this is approximately normal but now there's a slight difference with names about since I the mean being the means about how long and the variance with larger for it's a wise the variance larger will there is a lot of they'd have society has a real problem in Nigeria on both can be baked and therefore usual the becomes you size of the 1 always square to salute square them and then they contribute 50 50 but this guy is always going to be a real number so therefore you get this variance being slightly larger I'll explain the meaning In just and similarly 1 can connect if you look at the logbook I'll have said it takes a model farm and I form on the canal have to submit so here we should be careful not to look at the overall but maybe with the fundamentalist commands to act for which the sign of the functional aggression is positive it's a sign of things to these values are also hurled them and there's no more to be said then I also alters a conviction that this is approximately With me my my times log and so and then I I what I want to say is that if you do the analog of the calculation that for the for the functions the took a moment in the trial of these families and reunite them in terms of value being offset size on the news these 2 conjecture you will produce exactly the exponents gave escapist 1 or 2 in the syntactic case of torque a different and Cape Times came as 1 of the 2 and you can kind of see the difference coming in in the means which is start small in 1 case wanting the name of negative the direction where the powers of the little moments of a smaller Katyn came minus 1 or 2 and large the India so let me give you the kind of a heuristic explanation for where the central appear come from the state some recent results on on the subprime loans for so let's so maybe to start with you can think of logs laid off after psyche you go you can moralist think of this as the sum of all kinds of money really helpful like of course income but it's not it's not very far from the truth if you can get that some at the height of the the it's kind of roughly very good approximation of maybe we can get some that exists on the power of the world break and then this is usually a good approximation 2 I think this is not completely clear but it's real let's pretend that something like this system but this is not complete because people 1 more easily if you take although was a function it's not just the some all kinds of some or all the kind powers with the formal work gave Peter so we have to worry about that the plains where the kind you don't the prime accusing the prime the fault at all the relevant because they're confident the facilities when I'm
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on the halflife but the prime Square could still be slightly problematic maybe it would be something like a sum of 1 or more people want us to wake In this function is really doesn't .period effective learning much you can kind of say that this is never going to be so it's a bit like the problem of L 1 kind of hell so all the log of 1 kind is like essentially almost surely this is convergence and if you like you can forget about forget about the contribution from the French quest so when you're left with trying to think about this summer Prime sucked up to acts of people who have authority and I found here you want to make use of the fact that the speed of the for different crimes behave like independent wearables as Steve so you can justify the least effects is not very competitive when you can for example you can compute moments of the subject so the effects of selecting 1 or 100 then you can compute the 1st 100 moments of and you know that you know how the distribution To this like some of independent random variables then that's the central appearance will apply to the it said the idea would be that the gentleman some of whole the 2 halves of the going price and so if the people is just beginning randomly the moralists cancel out and then you should compute the variance of I should say that but now if you want to do it properly for longer moderator I should take the real party should take the you out here and the computer compares the moment so this what this will give a kind of heuristic justification for the war and you can make this precise in the key aspect because this initial approximation that I down can actually be made precise if you're willing to average overtime so you know this could be a 0 on the B minus infinity and the formula doesn't make sense it holds true in sense of mine but now imagine that let's imagine that can make the same argument for let's say logo for l have "quotation mark quantitatively characters so in this case I would try to formally right down something like some people to some .period what kind people close to the but they also have some old friends the prime cubes and so on that's before what you can see that now the prime square terms are actually determined explicitly they behave there always equal to 1 and the peoples up 2 I said that it should go to some of maybe some power but this grows like a long long the next 1 and long blond growing source who is going so slowly that it really doesn't matter where container as long as some power effects or even I don't even have to be a part of it could be like Exelon blog Exelon square and still be OK this doesn't change very much so it's this feature that becomes the means of distribution which is about half a long block grants in the contribution for the crimes still began roughly handling the pluses and minuses taken equally often so this should have a normal distribution this part should be normal which means and variance about the interview some of the principles of the kind which is about 1 again because of the thought the logic speech doesn't matter where you just as long as it's remotely sensible of the community but if that's the keeping state conjecture for the family of but I think there's a mail functions let me just explain the end of this calculation for the functions of because of Of the model so if you're right if you like your model probable thanks be to appear multiply 1 and that the people data in its quest to save then if you if you if you the answer to this calculation of loss and have a by but if you do have to evaluate the logarithms of these quantities to get out of here all or you have be appealed to the health and the health of the square was been peace so this fall like some peak the high terms which I did so this again we would expect so there will be multiplied by by a kite and this will be multiplied by Square which is just 1 so against these some fixed numbers whatever they are these of a peace but if you if you we would expect that this is not normal with me in the world and the variance was going to be the summer AP square overlooking the square feet but by rankandfile would then this would be his truncated at any reasonable .period is still be a component but if you know if you look at so you can check that out for peace "quotation mark possibilities "quotation mark but what would be the coefficients of Sematech square function attached to it and that averages out to the world the languages of the world which means that the peace quite possibly the square averages to minus 1 almost always so this averages to minus have time for and so I mean shipped in this example something negative OK so also a conjecture In this
54:14
context and what I have said that if some uniform emotional these conjectures was true then you would get compliments of the right order of magnitude 4 moments in these in these various examples so let me and my but some results from on upper bound which only described tomorrow so 1st there so the principle of law bonds was that if you can evaluate some plus a little bit then you can get more for all the all the higher moments so that a complementary principles which madam I have been working on recently Is that fair so instead of roughly as if you know some moment of steps along again the current alone the illustrate this by wealthier that in the in the context of of a model form or not to come to you hear the only moment that we know how to calculate is the 1st this been asymptotic sport and it's only moment that's the unconditionally and so what work is that it from 1 can prove you get the right of all apart from that we have a pretty good
56:41
results if we note but there is also this problem has now recently been but fully understood if we know half our age so some years back I proved that all year 1 gets almost constantly so for example if you would get but I doubt that took a moment of data assuming our age would be bound by the look into the case but that's a lot for 4 and last year there has been a beautiful refinement of the by heart what exactly show the key to the consequence of the upset so in some ways we essentially an assembly the size of of these moments so that they do you may be 1 more example of this result with Max you could 1 gets that they'd have to clear his pond .period he looked into his quest for for all careless and which was previously known if you assume that hypothesis but this result unconditional that we have more very often and we also have good upper bounds especially almost all the time if you're willing to assume during copies of the gene for the but that the idea would be that if you assume duration for 1 family or working with you get you get the correspondence under the last thing that we discussed these results tomorrow along with them up with 1 of the refinement dimension these kids they conjectures on on long normality of this work with with with with the would also give you onesided bounced watch these policies become who we can prove an upper bound over for the frequency of flights values which is exactly the kind of Jennifer abound but we can't produce a correspondent thank you so they were very good and we found problems in 1 of the people of the a year so that for example you know I've proved that seveneighths of the selfinterest of the right but would have won 8 of them was then there would be no but no you don't know what you are on our accelerated beauty please with 15 16 but not 1 right to OK so there is more work of 1 result I could mention that direction which is due to Bob but if you assume the pots and you also assume that their rules which are in the central point in the sense of density that the number of times together close to have densities then you can prove the hour of about but you need more than that what is the security the of day this is the the of them this sort of thing .period the goal while it is too hard or too to that and dimensions 1 of the orginal calculations the Bush was their condition where they were looking at a lot of the alleges but by taking the oil product from I'm trying to make a conjecture for I tried to make the Bush was directed at but it turns out that if you take the on a product at some point which is kind of like taking the side of the logarithm right and and if you do that then you're by a constant on the constant like squared to yeah which comes from comes from the from the kind of and it was the we're going to deal with the soldiers involved in the plot is this In this through but was I think a lot of people tried to do something unconditionally we don't know you know we don't know so it it's an interesting question but we don't know but we don't know how to get anything fall the whole point moment or anything like that before so onto the moment this kind of thing that you suppose I ask you for the measure office said on which they will then be right unconditionally so I don't tell if I told you so but which says that this is going to be bounded by essentially keep time something exponential into the mind of the square something bright unconditionally the only thing that we know about the hazardous upon which you will be if we lost so when these incidents more industry get but about what is in uniform in a wide range of of this this is the only thing that you know which is the 4th fulfillment the content of the 4th so there is nothing that we know which would be the logical estimate if we knew that we we would actually have consequences for the gap prime central candles not bonded not the counterfeiting problem but every shot and the ball containing antique price yes there's a range in which often say if we used to get the onesixth of something that we know that measures 0 right right but there is there's some range maybe that's the stated . 1 3 of them and think would be so these 2 . 1 3 and thank God we don't have a good estimate of what this measure so it doesn't have to be enlargement that we don't know how to handle appeals for small which we still he that's the end of last year and I was in Oklahoma City in moments of Libyan leader and the only 1 which is on the east and west and moments in the 1st year of the contract is worth and analysts said the it's not something conditioning isn't that interesting ?question mark locations in the north of on that would that would fold wood but would not follow through on the phone for the moment is that while rival nor of the moment for example part for which all but he will hold talks on the status of their apartment but actually like to make precise control which would buy them really yes there too just that 1 might as well say problems we accused of having the final say likened it to the of the he looked into the case square you that's a bit of a a million billions and uniformly in all of it was agreed with the inadmissibility of Paris Convention right it agrees with the appearance compared to what I thought of my opinion I think what you know what I think it might actually be the point is that the concerns that from here are actually very small so this is the factory reconditioned the defense would if you knew this uniformly in all capable applied the lawfulness of their all but you can yet but you can check that the formula satisfies it's not what was the 1st the notes of the the New Light of the World Cup for the 1st time in history what he wants to assume a uniformly for all so for example if you take this very small moments and you will be controlling the size of so by looking at the various small moments dedicated as a unit cost of experiment on the surface it's a bit hard to say what can be done .period reasonable answer Connecticut I think the question the move follows it's appropriate that the complex moment for me and it can be done but for all people normally used to make a statement about the problems of world was believed also so this is something which is worked out in his Houston PC and but 2 of the idea is that if I do that I'm doing this from memories of the I'm not completely sure they've really is so that's a really is like GOP minus Key and you looking at values of the adultsize that I won a lot of money then in that case I think that there's a flip from each of the miners be squared away along looking just eat the those things overlooking like to minus the squared divided by something it just flips to having something of entirely reminds speed on the this most of you know we don't believe that the about but this is simply because you get very close to those on both values are much larger than say if you want me to minors the overlooked of the office of of the world have a lot of something like that
00:00
Logical constant
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Multiplication sign
Moment (mathematics)
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03:22
Flock (web browser)
Multiplication sign
Similarity (geometry)
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Order of magnitude
Power (physics)
Number
Fraction (mathematics)
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13:45
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Range (statistics)
1 (number)
Insertion loss
Water vapor
Parameter (computer programming)
Inequality (mathematics)
Mereology
Number
Power (physics)
19 (number)
Goodness of fit
Sign (mathematics)
Meeting/Interview
Term (mathematics)
Square number
Units of measurement
Standard deviation
Product (category theory)
Rational number
Forcing (mathematics)
Projective plane
Exponentiation
Physical law
Moment (mathematics)
Bound state
Thermal expansion
Staff (military)
Nominal number
Functional (mathematics)
Sequence
Approximation
Equivalence relation
Hand fan
Connected space
Field extension
Formal power series
Summation
Voting
Lattice (order)
Right angle
Family
Resultant
29:56
State of matter
Multiplication sign
Connectivity (graph theory)
Real number
Direction (geometry)
Range (statistics)
Distribution (mathematics)
Insertion loss
Distance
Mereology
Power (physics)
Number
Frequency
Mathematics
Sign (mathematics)
Population density
Meeting/Interview
Lecture/Conference
Complex number
Term (mathematics)
Torque
Analogy
Square number
Subtraction
Physical system
Area
Logarithm
Physical law
Moment (mathematics)
Neighbourhood (graph theory)
Exponentiation
Variance
Calculus
Set (mathematics)
Mathematical model
Variable (mathematics)
Functional (mathematics)
Measurement
Approximation
Connected space
Calculation
Arithmetic mean
Summation
Voting
Normal (geometry)
Fundamental theorem of algebra
46:06
Group action
State of matter
Distribution (mathematics)
Insertion loss
Parameter (computer programming)
Mereology
Order of magnitude
Number
Power (physics)
Prime ideal
Frequency
Mathematics
Lecture/Conference
Term (mathematics)
Average
Wellformed formula
Square number
Negative number
Game theory
Logarithm
Block (periodic table)
Moment (mathematics)
Normal distribution
Physical law
Variance
Independence (probability theory)
Infinity
Mathematical model
Functional (mathematics)
Bilinear form
Approximation
Calculation
Summation
Arithmetic mean
Logic
Cube
Heuristic
Right angle
Coefficient
Directed graph
56:39
Point (geometry)
Correspondence (mathematics)
Multiplication sign
Direction (geometry)
Range (statistics)
Kontraktion <Mathematik>
Mereology
Rule of inference
Hypothesis
Number
Prime ideal
Frequency
Centralizer and normalizer
Population density
Manysorted logic
Lecture/Conference
Wellformed formula
Square number
Units of measurement
Condition number
Logarithm
Product (category theory)
Surface
Moment (mathematics)
Physical law
Exponentiation
Bound state
Content (media)
Incidence algebra
Measurement
Estimator
Calculation
Hausdorff dimension
Factory (trading post)
Normal (geometry)
Right angle
Uniform space
Family
Resultant