2/3 The pretentious approach to analytic number theory

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2/3 The pretentious approach to analytic number theory
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more In the past the more In and yesterday
I II about competition having for the triangle inequality that progress will be about the simple book about again today to confine the nicest proof and that was a nice person to me last nite and they have a problem I I remind me of another competition will sounds lecture also comes in the Americans here will know when talking about many of you may know of public competition 1 these competitions where people have sponsored to and with the questions in the time period which show their and national competition in North America and some fun questions and I happened to be chair of the committee studying examined 2010 and I 1 of the gold in all of these everywhere in the world is you put the year into 1 of the questions raised and we really struggling as a committee find a question with 2002 in there and this is just soon after all these conjectures sampled about that moment being named the Hampton post-up lengthening the time who isn't all these moment conjectures and I said to Nathan you go hell I can say is that the question you can think of leads 2002 and the came back with well it's this divided by 12 you try and find a problem this is true for the partners had we had fun for 2 days but it was always going to be a little too hard for the public and that's the competition find a great problems involving the moment gets 2000 the effect of the littlest so yesterday we've got to look at it and some good I told you that so-called pretentious or age directions for roles this report has grown from 1 for all the the case derivatives factorial it is less than less than something depending on how to handle but just give some explicit say 1 close to the Epsilon was signal In case of conjecture could be any constant depends on and which saw yes they supplies place 3 officers and I claim was starting point for for what we're going to do so clashes quickly run through Peron's formula which we yesterday so that the but there are serious flaws some of and around S so I really have to do is read off the coefficients 2 interpreters for water the coefficients To some analysts to acts of the coefficients well .period approximately 1 2 the integral just the right ones someone wants see 1 he asked and so on OK thank the prime overstates the coefficient is minor slander and compromise on this side there's a minus a finalization and efficient against coefficients here of mine is a paean to officially minus 1 and this only has 1 coefficient which is right at the start so what we can do is to guarantee approximately well this is 5 x 9 effects and tell you well and good to think of 100 minus 1 using the sort of balancing act in Green use of off main terminal the author authority what the and organized I do want to next year or in the case so that we want to play the game case derivatives so what happens when I take a case the relative when I've got their 1st series of the summer a endeavoring to be and I take take the case derivatives of respect U.S. then all really happens is lot I just get my lord and each time I take a direct so action if I just put a here if the relative there's a plus or a minus I'm not really concerned which and I've got a log into the car OK years close to talk a lot I'm sure you know you can go between and pine 1 thing about the counter Prime's very easily by partial summation because loggers is very smooth and benign function and long as this moment isn't incredibly high is often make a lot of different styles of Outlander minus 111 reminds sometimes Logan so OK so let's do what we did last time which is just a constant values and see what happens that's right hand side and less and less than remember 6 years 1 of 1 of the log which allows them to save it from the acts as a graduate the which is the so that would use this formula we assume that this is assuming the potential treatment of and we get will forget the constant so we get some sort of integral between minus the and so we take me change over the society let us time we've got to to the can a factorial that into 1 of the founders of the lessons could leaving before the Epsilon 1 I've got my acts of that outside got this 1 in the the so that's another abound now let's disintegrated integrated but this part and that's when you integrate just going to the Epsilon the treaty to
be someone's 1 and nothing bad happens New equals 0 so this is less than less than 6 times the Epsilon times due to the KK factorial which is essentially to to the K R I want to that had no time to replace that will go into the teeth of a lot of things OK and there I got probably to take account of Europe and used correctly there should be a lot of them just as this worry too much so what we want to do OK so we're eventually going to divide it up among the partial summation of the divide through biologics so what want to do this is what we get from partial summation is something like and maybe a few logs that occur because of partial formation but we'll get something like that to the Epsilon To logic that not comes from here and then all we have to do is make the optimal choice for the and the optimal choice to K will be pays logic yeah it was optimized things that when I take a close look at the 2 to forget that very faithful logic to when I get less than less than X 2 trips 1 and at the time of the year absence I take the heed be around that still something and we again proved well prove director announced that address that making assumptions about the function implies agreement the same estimate the effrontery agreement hypothesis we saw last time and when I did say last time is that actually supposed to of going about it in the interest of the next half steps on just choosing his Texas something OK so that's used the potential free pulses recall 2 prove very strong them on my ex and now but let's just see what happens with this if we plug in a different manner that's what we don't approve instead of this is something like log for 1 thing some constant but 1 of the and a unmentioned in today so let that sexual start from reminding ourselves what we did initially to proof of the shows 3 hypothesis conjecture if we replace now by this system what will get unconditionally another sketched out in minute then what we have said that the 0 officers when we look at it said that we looked at said that close to to an era then what's the radius of convergence of that events that we can get if we make the assumption that the derivatives abounded that quite well again and then again terms that have less money says 0 a times this thing right this is is going to converge it something like this might hero is less than constant log 1 the same idea of a 2 to the case and we have that and so in particular this gives his era free region this is exactly the classical 0 3 regions of the person that comes out of proving a little more transparent didn't get to go through proved yesterday of the regiment you go through it it's little how I'm used to it but it will prove this is having a lot easier to see wife their 4 regions just yet series of the convergence people so if we plug that estimate what we do is reserved for region and then proved the Reuschel prime number theory or we could just play with and here and there and the only real changes that we change that To see log 1 plus the 2 became and so when we come down here we have Hey a kind of C log 1 Over the log next week of the manifold just a place to buy the log on and then the optimal things will be OK as lot acts overseas lost 1 of the and so we end up with less than 1 expert the 1 minus some constant over long it's something that we have careful America should what have be we have a 2nd error term the prime number theorem that's flexibility maybe stands and housing log you as normal you just optimized for the 170 is the same as the ecstasy of along just like in the classical proved and that the optimization occurred along the lines and so you end up with the upper bound next to the theory of a case that recovery of the classical 1 year old the proof of the 10 firms should about and there yes sir the issue To improve a prime number theory the Riviera town the Panama fairness to improve disband if we can all
lecture and should prove this the question on authorities have lost control of his trip OK so how might we go about proving such about so we need bounds on Z was remember what got up there a prime plus the 2 months get former take case derivatives at best then we're going to have the case relatives privatization K times plus became derivatives data so these are things were going bounds on this presumably a straightforward because this guy is just the some of what we talk about this a minute from suggesting trivial it's a it's doable because the plus-minus His that's the case through the brings adventure and we were always the right 1 and this 1 is more tricky would you do that well we just differentiates it from day to get what state upon privatization minus Sopranos April square you do it again and do it again this uncommon for mass and we just so happens that that's been sorted out for us here and that this is say I guess tutor was fired the Brown Root so if you find a big mess like this ,comma Tareq spoken they claim it is a common foreign institution from 730 if you find this in a in a computer science book against their editors from 1987 so we believe common on Bruno's problem we try to hide abroad I'm supposed to know you're going speak so languages Zeta privatization the bagel common format times something involving Zeta differentiated a set number times was a but the way to look at this and let me just note that this is this is this ,comma tourists and you can have fun with it and you can find out that this whole some actually is less than an absolute value to to duplicate minus 1 the action lived for us this is all we really care about K Line is 1 times the maximum they take absolute about so you just have to take the was summoned to me every time back OK so we're going to have to work with a lot of terms like Zeta head peseta the jails Over the past so what we we need out of bounds for these guys that was a forgery and we about on upper bound for 1 of his 8 arrests which is the lower bound for the West as you might imagine he's a fairly standard things in literature this is particularly to the right of 1 writer 1 convergences is on your side but let's just take a brief look at what might be difficult so if I you want to just stand this up and I'm thinking about sp Bonsignore IT and against it's not very convergence of signals just to the right of 101 of some guy up overall interviews and then the obvious way to do that is to say Well this is roughly the integral right we prospects sums by intervals even complex so we and we have to be with it if it's real this is an easy approximations of the arms it is complex that's a little harder but essentially you want something if this is going to be a corrected consummation you want things to go smoothly now let's just forget the signal for a minute and look at the end the IT what happens saves you go between 2 to the IT and industry the IT things are going to be soothed there's roughly the same angle involves yet With the gradation by angle is move from 1 incident that as much if it around the circle many many times then who knows where it ends up being covered approximately integral so if the they tend to be ITN and was nowhere near 3 the eye the thousands of the IT is very close to a thousand once the reality if you think about it and will adjust from end to end this 1 was just compared the angles and those 1 the IT Over into the IT look something like 1 plus what I hear over and if you look at right so where things start to get a headache is 1 end is the give-and-take an event is less than the things the knowledge this is not a good approximation essentially we have to break up into 2 parts we have to say this is like the sun for many equals 1 2 and 1 over enthusiasts I'm lost and some great all 1 of the way and save the world from the state of course and then this guy without a doubt looks something like the the integral to you be asked Dr. that's easy and this 1 is a mess and the best we can do really is take absolute value so you get 1 over and for the sick so big 0 yeah so this is a problem and I there are different ways of coping with this and we just saying moment that the long distance fairly benign Sigma grief 1 but now when I put on here log into the cafe the same arguments apply when I put I love you so the case but now this time is going to be something somewhat significant and if you think about scenery Costa won the sum of for the care and makes it look like a log in to the cave was 1 and the other directional corporate smaller the company in a year's an approximation you gotta figure out ways but everything else behaves this doesn't back so there are different ways of coping with the US talking I guess the 1st question is is this a good I think to put into full well you bound in the case derivative it's actually look into the K would be more convenient distraction quite true but that's what the sort of thing you're aiming for so if we put a lot into the plus 1 into a formula we get something here like a lot of the 4 To the instability of folk to better than you can get something but it's not as good as the cost of the free region so In the beaches this paper he has the idea that 1 way to get rid the deal systems to get rid of it With so what thought it the last time 7 regret this patient announced wise the exact opposite of what it was last time put bracketed placenta now that nobody bracket rather there is a significant step so angered who won divide and placed his greater than what and here we don't need statewide very big it's just like a parable log notes in With the square like so we can work instead of with data women workers removed a small prime I guess and sounds like she talked about the pain of small problems 71 has the pain of the small print but you know the small prosody when they last example we've taken from each of the Rolodex and that's what we chose me and others yet so those aren't very big crimes were removing for counting the prince of wax removing the primes at eatery logic circuits so if we go back to Peron's former everything else and just deleted the small prime it's something been really affected which should just be but here it means that this guy gone now the problem is that once you do that this is not so simple because you know something overall and you have save by the prime less than or equal to and unsavory computers and serve estimates was on a big deal you can do it fairly standard techniques the majors were basically
the proof if you could do better estimates for the case through of Zeta then you could do better let me just formulated by version which annuity John but this is fun and incorrect and so I if you're trying to understand this summer this sort of some as I said it's a signal sighting in the thing that's interesting to me that if you like from partial summation the log and the parents secretly financed from reasonable size is unimportant the 1 around causing the cancelation of interest and the estimate from Leningrad which turned out to be used in Aqaba but it also can be used if not just as the black box Is that the sum of inferiority and the 2 acts on the mean value like there's less and less than exponential might say in Logar look for stand alone this allows you to what you can imagine his partial permission to go through the estimates that allows you to put a two-thirds in maybe this log floating around you can put a two-thirds in here but you can see the two-thirds rearing up from the log excuse of square when you put the 2 thirds in here and you go through the the calculation overkill that is what we have we're going to end up with two-thirds here the and then with equating log where at loggerheads over long before the two-thirds of the Pfeiffer and so you get clause so once you've done this give the students and the IT estimate is about boxing you can just go ahead and get sample from them but there were a futile more precisely get a lot of 150 in the denominator so you exactly vinaigrette Carbone so obvious me lies in the estimate for Bruce transformed from them but this is not something easier approved but it doesn't use analytic continuation of the function of the One it does achieve our goal of getting stronger all the prime number therapy without venturing to the left 1 this is a no 1 of the so I'm is a 1st step but knew little more you can get a lower bound status simply by latest Sigma data signaled society 4 the latest signal Fig again used batons identity and so since we have upper bounds on stage rights of what we have a lower bound on on status as a consequence and I think quite sharp enough to you but if you start doing it would remove small prime factors the case so let me go back to the distance function and yeah so I'm in some sense is a little disappointing so the sounds Britain's argument is begins to feel like a pretentious proof of not only the problem with the of strongest named formed from them but there are lots of information about a clever ideas and you end up the same result examined such a weird result log X the 350 Loblolly it's the 1 for what reason why is obvious because the deep idea that's needed to get there is sustained deep idea I this that tells you something about me mean I haven't really said this before and now negative about what the history of working on prime numbers from the perspective remains at a function is what is it that were really doing 1 week but proves 0 3 regions in the classical way so but you could argue that special fuel could prove to be the argue that His one-liner is is for region the remains function that we know of and so the major sustained the free pointed out that the city is proof so here we sort of avoided going across the one-line although we can know note that gets in there for a region 1 of the classical proves is in a sense what they do is they say we know there's a function so well to the right of 1 that in order to know just to the left of what we can surely extrapolate and we have to really clever to get as far as we can go that's roughly so sadly most of about 0 4 regions of the shattered Henrik is extrapolation to what happens the right 1 so there is some logic to never going to the left 1 because most the consequences of more elementary but guess no at the end of the day this new approach if you like which isn't new but writing it is 1 of the whole thing is perhaps you and the task approaches already equivalent many many ways but there are some advantages to the new approach returned in triumph .period so let's go back to defining the distance function molded over here therefore distance function and be interested in I gave more general definition so the f and g defanging multiplicative they live inside the unit circle and will define the distance between Earth and G with respect to limit 6 years square is some over the prints up to it but 1 minus a real part of In the proof of the tremendous direction what we what we talked about yesterday is on par today is that the meat of the prime number theorem lies in integrating Zeta Prime Minister of State us and understanding that as we will times ex-US arrests and understanding that and as we just still working as a surprise is not that difficult coming pretty well understood functions of some of the log in and around the essence of some feel especially the right 1 so this is the element of fear here is not promises not actually has not the 1 of his her so you can actually show that getting it estimates that is equivalent to getting good estimates to 1 of his interests their elementary ways to to do this but maybe not given at 1 of his major actually restaurants and then if we used from former again willful 1 2 iPod which amiss but you end up with some of the movies function to so in fact the prime number theory is equivalent to the same statement that the sum of the movies functions the flow so square free integers are as likely have an even number prime factors in all Member prime factors and in fact the balance on you and you get is exactly the ban would be doing with the prime minister analytically much of no I like you and because new event it is function and that leads into a whole different fields of things we can work and so the question 1 might ask is not just 1 is misdemeanor value of the sun and 1 might interpret the meaning that if you have an little 1 attends the 0 but you can
say announced that the larger context which is 1 of the value of any of the functions going to 0 convening of functions mean value is not that lives inside the unit circle more than 1 and so the IT thank you so what is a good 1 that doesn't have the value their but in the IT it is a great 1 forget everything I said before about this difficult but it stopped because it's not relevant this is more or less but we will mean values 1 over access is moreover unless 1 over X times integral of you to the ITT between 0 next and this is this is just a equal 1 over X Dexter wonderful sightsee overwhelmed society and then when I divide by X together because you taking notes or give you 2nd hysterics it's something worth thinking through the use of the thank you it's also worth getting correct the package you this OK so this is kind of a coalmine Valley because look if I take the absolute value of X the IT I want society it is what 1 word 150 squared some number so when I take the mean value of enter the IT it doesn't tend to 0 the fact that the value Kansas and constant begins 0 but would useful here is it anything the mean value right elects process goes round around the circle so when a little bit of a strange place mean values Mulford functions circle because the necessary handful anyway this is another example of examples as from friendly real that the mean value does not deserve a good many other examples the answer is yes but so what if I took 1 1 was something that goes across from us at equalizer but instead I took half of the government BF appear to be 1 on every prime except 70 and on 17 I took half of Peter F. 72 behalf so when I think value thing it's going be some consonant with the constant but could you not going to be 0 it's going to be something all the time to the reduced size almost all the terms you square and on son so it's not hard to find other examples where the mean value does not good to 0 but the mean value but those examples are taking the nice pretty examples and mess with them that's reports what is the limit of how much you can mess with pretty examples of and the and this is what bolstered the distance coming in minute but you can ask Are there any other examples of the end the IT and the mass of them a little bit but doesn't really change it just changes Conte are there any other examples in the great fear of policy no of examples of it says that if policy I guess it's the vibrations at states that year Joe FN multiplicative FMS to 1 day after the mean value of that not then f is pretty close to being an end for the IT and what is pretty close to me then there exists the real should the bounded even such that alone the distance between F an the IT going all out to infinity bounded the last unless someone unless infinitely and all so this is what we mean pretentiousness that pretends to be the end of the IT maybe and that he may be close to you know we use the word is approximately but that the potential there was something that was more surprising when we initially calling the phrase 2 things really have no right to OK so that is what happens to general multiplicative function and the question the prime number theorem is from you event so we send a the prime number is here in the fall it would be the same thing as the some of the UN those knockers 0 which by Hamas experiment applies that you pretend to be and so the IT which as we said last time from the triangle inequality this tells us that we can just take square system you sweat is 1 into the 2 like he pretends to be 1 and that would apply for instance what many ways to prevent incidents but 1 way to prove just proves to that data has a poll of 1 2 so because the light sit stand to approve the prime number Themba nicely the classical proof of but he is the great thing when you are dealing with the classical proof of prime number theorem you using Zeta and its properties analytic continuation that its analytic everywhere but there's a real structure going on with you don't have to worry about structures and multiplicative function the same notion of pretentiousness is all simple it allows you in the same class a kind of questions to generalize vastly from coefficient of the series that as Taylor series 2 Place series that they necessarily have to be world and that ends up being an enormous difference in what you got to prove so but many meets say a little bit more about this distance function so let's have a look at this was the 1st serious so let's suppose that after and you know I said the last time that when we do a problem of accounting things up to acts were not really concerned about the value of the functions Prime's beyond so when we look at some of the and to act which we can play with the values of a friend greater than that so let me assume the death of Peter is 0 yes he is grateful Max and what I want to look at is effervescence so the sum of that thin over insidious announce a multiplicative function not like here we're taking arbitrary directly series and we're looking at the product of 1 one-plus FPO repeats of ER plus 50 square feet federal and but this is only over the primes up to it and so that that gives us something nice and finite supply worth and when we do our integration where we're working where having the real SSC the society and C is greater than 1 you I want to understand the size of this just like exams lecture the terms after the first one role at 1 of the Peace squared they converge so up to a constant and not relevant to the size so up to account this looks site exponential all this summer you have to acts of Over pts and I've written this is p the C IT looks like that and that's really just FPM PVS rewritten then he
is off the coast of 1 the change saved 1 but actions that cheap that's allowed the pontiff 1 of the log acts again it changes things only by constant now this is a bit like when we did the distance fees 8 arrests in the press 0 I'm going to add in if you like a S 0 what I'm going to look at is actually something weird have sought officer put this is a minor and a minus and finally won a game I'm going to add another test which is on and add 1 minus F P repeatedly before my anxiety the the so this action ,comma managers of minus the sum of 1 overpaid which is a lot of acts on an expansion to acts put Lauritzen their this up to a constant you can see where I'm headed and so now let's take absolute values revenue so that the society if we have the values what happens after values we take the real experts the real part of it report 1 of these the group of 2nd but not so easy to this is log at times exponential all of 1 mind real part of the the United was exactly what's OK so this very much that this distance function it's got to be relevant to understanding these integral sprung for someone we look formula for integrating up line just to the right of 1 of the best that's right and as we said our approaches just take absolute values in the most moronic way try and understand will put to it seemed to make it a less moronic but the key is going to be understanding the size of as we go about life and the size of while here we have very good guy so what's the maximum size of Africa B it's going to be reflected by the minimum that the distance from but this much like what alliances because they're talking about minimum distance right there exists an end there's a such that this will be the minimum distance if you like so actually just realize that in the audience that I should say about I took a lot prove that if you looked at the book book of venison Kowalski sensible if same and that was the inspiration if you look at it as an elementary proof of the crime and the fear of it was sort of a bit like this sort of elementary and users ideas zeta function and so on what you see there are many elements of what I talked about and :colon is proof that the origin of his proof was when he was opposed studies professor the symmetrical when he was a post-up we exactly talked about combining about his casket with potential and they were pretentious but not pretentious he was all pretentiously this is the last and it's central to the subject of of the promise Hopper's proof policies and also slightly edited by sound and loss of the board when I guess I think it's all about to of southern brocades and a convenient way to figure it out With effort yet let me be a little more precise than about the storm sewer and after will be the maximum With the lesson recalled the Of the S C +plus IT so the maximum that their places gets announced value where considering the integral and as we said this is like exponential all mine and logic blogger 6 times the To the minus the minimum the the distance away from Clinton so that's a connection will be useful to have that Maloney died giving users the vague version of policies there let me give you the last version of the which is the uniform of abound on now again if you look over there you've got to look up there some and that take investment you go directly series actually s of so we would like acts what's expected the 1 Everest has probably been mine so the main thing is the value of their faces there is the biggest thing can get and Hollis confirmed in the action proves that some analysts is less and less than this maximum that their faces can be 1 of the world so I explained that the minutes and here we have to take some of the great logic there are other versions of the terrible images state research and explicit way of doing the same thing when issues said about something's wrong I'm sorry I meant to I'm sorry my notes brilliantly written which help to you but let me do this natural to divide through biologics should and this is less than less than 1 might so it's a matter faces about biologics logic is the maximum to be with you can see that the mind square so it's founded this maximum
and log against social production values and loggers negative so if you might suppose this is 1 of a thousand because that's what this is pretty small it's the 1 over a thousand you care about focuses on that you get the maximum their face serious divided by law there's a little bit extra which you can concoct an example to get a little bit extra probably about the planned to do about possibly playing today this should tell you some nice examples of of promises that I have on May to limited wanting to it comes perhaps the most of this year some of natural examples of of multiple functions that have been worked out in literature like you and an end the IT but this 1 is particularly lovely and it generalizes UN but in a surprise ways of and is the multiplicative function where he is minus-1 and all crimes I'm not worry much about the peace squared because they they can't find your account and really affect me the issue so a generalization would be made after the Alpha all or primes 10 rationalization and if you do that well what their places on the work wear size said that working the what happens on the prime square thank you doesn't really affect things very much so you wanna pick their easy their faces as possible what their faces is really easy and the sense that we are familiar with that which is has at every Zeta the Balfour so they will affect the rest is a the alpha then every F equals all right so this is the subject of I think we should work by itself the the referee's report on such as paper and this gives you which was subsequently published that the value of this settlement will from every price looks like and I'm not going to give you every single detail but argued that the important thing see over gamma also the little 1 time text block access the office of my life and the health is I well-behaved those words this OK so what verified in some cases we know L 4 equals 1 at pretty good luck 0 2 1 and constant constant we know the constant 1 of these Alpha equals 0 the more functions 0 everywhere the terms of the log minus 1 that's not retaliation that multiple functions 0 actually 0 so this has been here began 0 is Is that what know about gamma Gamez metamorphic and has a poll at all frequencies 0 minus 1 and minus 2 that section so this also accounts for the view the prime number theory right if we take efforts minus 1 it appears minus 1 wheels get this is 0 and the things around so you got to balance here at the very special points where the man is 0 my financial minds to live outside the range of talking about you do get the mean value 0 at every other complex number it's a non-zero mean something on 0 times some taken have along this is how that compares how Lincoln's database yeah so I if you compare this to compensate for the Alpha and that's going to show them what we wanted to know how big this function and so it's going to be dangerous the health how they can that be member this is a trust will we truncated and acts the 2nd is because logic and for the Alpha overlord acts so hard lessons that gives you the logic the alpha minus 1 in the upper bound and it's just some texture which so how the system really does well and beautiful examples when you look at a book at some point you will find that you can concoct means of law blog by and playing with the example Alpha equals on and changing from went didn't into here there is an interesting question which says Wayne is harmless in the right ball when to give you know about the rebel because we know 2 examples 1 it's nowhere close the recounted the mean value is much smaller effect is always 0 mean values much more and much more interesting of P is always minus 1 the case the movies function the mean values was much smaller than the of production so that is unit circle come Is it is some way to predict 1 when this kind of fails to appear in another achievement of reports which quoted could properly I'm going to have to made technical Congo so if you like obviously report greater than or equal to minus 1 for the size of this thing is bigger than logic minus to so what it says something about the following year for or if consistently uniformly you have something like the mean value His last nite and would like the logic to the 2 . 1 3 or tell exactly 10 but 2 protect prospect almost as good as possible and then f pretends to a certain sense it's very similar to that To be either 0 a lease on average Wolf the UN and severity so yet the only way you can sort of get a moment value offered function to consistently not because it's the right answer is in the 1st special cases but we already know about OK so not good don't tell them and it's really ridiculous OK maybe I'll just finish with talking about the new proof of the triangle inequality with this function may come as a surprise shocked to hear that Terry e-mailed me last nite with the eye approved than we have yeah these guys live on the unit circle F and G then if you look at best minus that's a P minus 2 squared after balance then what is going to get you going to get a 1 from the aftermath of the square 1 and that's about GPS squared and my wife reported that the conjugated twice 1 ministry g Contador OK so when on the unit circle you can convert this into some sort of regular I think so we could define all of the for that goes on but what can best minds due to be sum of the 2 also squared to be 1 of best -minus cheap What is this look at that's exactly the same thing as the distance function so the norm of which is the same as that of function
and now we know we're comfortable with optional but we're playing on the circle call and we know such that you have to call and stage and he suggested things on the unit circle but of course with norms in this sort of thing you could in fact normal you have linearity so terrorist nice idea of which leaves him before was that if you do have a multiplicative function with a value of circle the prime then why not treat this as the average of To multiplicative functions that have values such blasphemy OK so you do this each PC as the jeep in HK he just average McCall is a probability of whatever you want your average over the and then you do exactly the he just these altogether garbage over and that ends up giving the triangle and called for the reasonably accurate description of your apartment but Finnish retirees this is not good you I want to know what was going on in the words of 1 1 of the things that you will be the last time it was the last thing you can do this too this is not the there's a I'm not really talking about the rich history all of this year member of the House of involvement in this beautiful articles and refocusing on where right now do what you want something honest and niece and 50 years of various 570 is 69 left with the things 67 the 63