 So, today we look at how to extend the zeta function to the entire complex plane and in order to do that we will take a detour short detour, but still we have to take a detour and introduce a familiar function to you which is the gamma function you know gamma right. It is the definition of gamma function of course, this is generalized to complex numbers. This is was originally defined for real numbers as with the zeta function and what we know about this function is if z is a positive integer then gamma z equals z factorial. So, it generalizes the factorial function and that is easy to see right just integrate this by parts. So, do you get z factorial or z minus factorial let us see you get z factorial you integrate t to the z minus 1 that is right. So, then that is you get t to the z by z good. So, now if you look at this function and think of z now as a complex number. So, what can we say? So, firstly we have to talk about this guy being first it is defined for positive integers very clearly according to this, but when z is not a positive integer it does not have such a nice expression for itself. So, we have to consider or analyze the behavior of this function on non integral values. So, first question that one ask is does it converge everywhere? So, what is your guess? Does it look like converging for all z? Actually it does not and it is easy to see that. So, let us just let us get back to this. So, this definition is of course, valid for any z that is how it is defined and let us integrate by parts. So, we get t to the z then minus 0 to infinity differential e to the minus t which makes this plus and you get t to the now this is as for t equals 0 this is 0 for t equals infinity also this is 0. So, that first term is vanishes away and we get 1 over z gamma z plus 1. So, actually this tells us so that gamma z is z minus 1 factorial. So, this but this relationship gamma z equals 1 over z times gamma z plus 1 holds for all z. Now, when z is 0 or gamma 1 is 1 that is straight away by the fact that it is n minus 1 factorial. So, gamma 0 is actually so it does not converge at z equals 0. So, that does imply that excuse me analyzing this function is can be little tricky. So, gamma 0. So, let us carefully analyze this and let me write gamma z again and the reason why we are suddenly spending time on gamma z will become apparent in a short while. So, z now takes any value in the complex plane. So, let us first draw this complex plane over here. So, what we know is that for every integral point it is well defined as a nicely defined value. Now, for any z such that absolute value of z is greater than 1 not absolute value again I am making mistake here real z is greater than 1. What can you say about the absolute value of gamma z? This is going to be bounded by t to the absolute value of t to the z minus 1 which because the z is in the exponent and real z is the complex part of z will vanish away any way in absolute term. So, this will be some t to the alpha where alpha is greater than 1 sorry greater than 0. E to the minus t is of course, c to the minus t is always positive in the absolute value this and this if you see will converge that is the value this is less than infinity why? Well firstly alpha is positive. So, near 0 this is close to 0 as you go away from 0 the value increases, but then e to the as t increases e to the minus t really starts dominating t to the alpha and it very rapidly takes it to 0. So, this is a sort of an informal argument, but one can make it more formal by just expand e to the minus t as a e to the minus t is power series not necessarily because this is going as still going to be. Yeah actually basically you show that beyond a certain t this more precise argument anyone of you have more precise argument for this why should this converge only integers yes sure gamma alpha plus 1 yes that is true, but what about what can we say about gamma alpha plus 1 sure I can say that this is absolute value of gamma alpha plus 1, but why is this bounded no alpha is not integer, but can we say probably yes is defined through this definition and, but I think you can say that because if you see as alpha increases the value of gamma will gamma alpha will increase because you the curve will always be above the curve for smaller alpha right you are multiplying this by t to the alpha which as long as alpha is of course greater than 0. Then it will always be above the curve and so the area under the curve will always be more than the area under the smaller alpha and therefore only for t greater than 1 fair enough, but between 0 and 1 this is anyway bounded. So, then you do not have to worry about it right. So, that together will imply that this is so that now makes the life somewhat better that at least it is well defined on the entire real line not only sorry not on the entire real line, but actually on the this half complex plane which is real z greater than 1. Now, what about the rest of the complex plane so this like again we this is a situation like the zeta function there is a half plane on which it is well defined and actually analytic this is also analytic it is analytic function of z because again integrand is analytic function of z and the second where for every value of t it is analytic function of z and the integral is fine it will take you make give it analytic. So, it is analytic on this real z greater than 1 and we again raise the question and we the reason why I am want to define it for the entire complex plane will become apparent very soon. So, let us define it to the left of the real z equals 1. Now, here unlike the zeta function the life is or the process of defining this is much just use this equation. So, this is valid for all z so which means if you want for example, gamma half that is gamma 3 by 2 divided by half. So, you got the definition of gamma at this point in fact, this is true for all points you can come so that is take this strip between 0 and 1 here you can define the gamma value in this strip by looking at the next strip between 1 and 2 and the gamma values there and doing the appropriate translation. Once you have the gamma values here you can define the gamma value on the previous strip between minus 1 and 0 just copying that and keep going back and this way you can extend it all the way except when of course, it diverges. So, when does it diverge does it diverge anywhere on the right side of this plane real z equals 1 it does not it is value it converges everywhere we have just seen. Does it diverge on any point between the 0 1 strip no because it is just pulls out the value from the p s 1 and this divide it uses this relationship and the only way it will diverge is when z equals 0. So, the first value on which it diverges starting from the right infinity is at z equals 0 except for z equals 0 what is gamma is minus 1 gamma minus 1 is gamma 0 divide by minus 1 and gamma 0 diverges. So, minus 1 is also infinity minus 2 minus 3 minus 4 essentially all negative integers including 0 is where it diverges and it does not diverge anywhere else. Therefore, gamma z is a meromorphic function on the entire complex plane with poles oh by the way why does it why am I calling it pole at what is the nature of gamma 0 it is gamma 1 by z. So, around therefore, has a simple pole pole of order 1 at z equals 0 at z gamma minus 1 that is gamma 0 divide by minus 1. So, it is really again the same simple pole we will get pulled in here and so and they will all work. So, no higher order poles occur anywhere each of the negative integer point you get simple poles. So, good. So, now we understand the gamma function in the entire complex plane. Now, let us connect it with the zeta function and that is very simple, but it is it is the connection is very simple, but the idea that yes there is such a connection is not real. So, let us again consider the definition of gamma function and let us do a change in variable no the other way around sorry t is nu. So, for t everywhere replace nu. So, then what do you get gamma z is what happens to n greater than 0. So, what happens to the limits this stays 0 to infinity n to the z minus 1 times u to the z minus 1 e to the minus nu and dt is nu. So, n to the z of course, this comes out of integral. Now, this guy is familiar 1 over n to the z this is one term in the zeta function summation. So, let us sum it over. So, this equation holds for every n in particular for every positive integer this equation holds. So, you sum over n greater than 1 no greater than 0 1 over n to the z. So, this gives you gamma z this is gamma sorry gamma this is eta function equals this sum. Now, what I am now going to do is interchange the integral with sum again I have to be careful here these are infinite sums and integrals, but because of the uniform convergence on the of the thing inside the integral inside I can do this. So, I will go ahead and do this and what is this? This is a familiar series this is geometric series with multiplier being e to the minus u there should be an e to the minus u on the top here. So, now, we get this relationship between the zeta function and the gamma function zeta times gamma is this integral on the right hand side. We know that gamma is defined over the entire complex plane. So, if we can show that the integral on the right is also defined over the entire complex plane then we are done we got the definition of 0 for the entire complex. So, let us look at the integral on the right hand side is it defined over the entire complex which means does it converge first of all over the entire complex it is unlikely because we just saw for the gamma function integral that when real z is less than 1 then z minus 1 the real part is less than 0 and then the convergence does not quite hold actually it does diverge. So, we cannot directly use this relationship to conclude that the integral on the right hand side does converge. Unfortunately, this is slightly different from the gamma function integral which allowed this beautiful relationship between gamma z and gamma z plus 1 which allowed us to extend it which no longer holds for this kind of integral. So, that is also a problem. So, just by looking at this relationship it is not clear how to extend it, but one can actually use such a relationship to extend. But for that one has to do a little more work or there are two ways that little more work can be done and both these ways were given by Riemann in this paper. One way is to start with this integral itself and do something to this integral on the right hand side and then transform it in a way that eventually it takes a form which we conclude is defined over the entire complex plane and then therefore conclude that the zeta function is defined over the entire complex plane. The second way is more interesting which in that it starts with a slightly different relationship between the gamma function and zeta function and that is one I will show you because that gives something even more better than the definition of zeta function over the entire complex plane. It also gives a relationship between different values of zeta function just like there is a relationship between different values of gamma function and that relationship will be very useful for us later on. So, let us adopt the second way. So, the second relationship so again we start with gamma function and this time we do a change of variable again but not in the way that so I should have done this right here. So, I will not consider gamma z but gamma z by 2. Now, replace t by its pi n square u. So, this is a funnier transformation and it is remarkable I mean it is these are things that you just wonder how did it come by magic or something but it will you see that later on it will fit in perfectly with our analysis. So, what happens then? Of course, limits remain the same of course, n is of course, 1 2 these are always positive integer pi n square u to the power z by 2 minus 1 e to the minus pi n square u and dt becomes pi n square. So, this gives us 0 to infinity pi n square to the power z by 2 right pi n square z by 2 minus 1 and then 1 pi n square coming from here u to the power z by 2 minus 1 e to the minus pi. Now, this was expanded out pi to the z by 2 n square to the z by 2 which is n to the z which is exactly the form we like. And again do the sum use the uniform convergence to push in the sum inside we get zeta z. Now, this sum which is inside analyze as the previous sum of that is but do it advance even faster than the previous sum. In fact, that is the reason why we will we chose this because of the there are some points where it previous integral was diverging we choose a much faster converging sum here. So, that that divergence can be eliminated. So, that is the philosophy of course, the question is exactly why this form because this is something which we can still analyze not as well as the previous geometric series but still it is amenable to analysis. So, let us look at this let us give it a name w u. One thing that you can immediately observe here is that this is symmetric minus see for whether n is positive or negative this takes the same value because there is an n square there and for n equals 0 this is 1. So, basically all that is taken care of here. Let us give small w u let us call this. Now, this sum is a fairly nice sum. Those of you who have tabled with some bit of Fourier analysis no does not matter we will do a bit of Fourier analysis here. So, it is really a detour and again I will again warn you that there are paths that I am going to ask me why somebody could think of this I do not know but the fact is that it has been discovered and here we are. Of course, to to demystify it a little bit I do not think anybody sat down in 5 minutes and in 5 minutes came up with all this analysis it required a lot of hard work I am sure a lot of different experimentation with different sums and finally identifying what is the right one. So, now we want to analyze this and in particular what I want to do is w u with and we let us define a function have I used the symbol f probably not and consider the Fourier transform of this. So, let me prove the following lemma just redo this I should consider this as a function of n not as a function of u. So, this is defined for different values of n and then the Fourier transform of this is f at m and still write f at n or let us write n there is something here I think is square root of u or 1 by square root of u. So, do not take my word for this when we derive this we will figure this out. So, what is the Fourier transform of this function anybody remember definition of Fourier transform function.