 it is just one simple trick that has to strike you then it becomes very clear. So, we have this h is a proper subgroup by the way I think I have yes I have made a mistake in stating of the g r h or the consequence of g r h when I say q square root x log square h this is not true the q is here square root x log square q x may be this is what stop you from proving it yes what. So, h is subgroup of that q star is the proper subgroup. So, let us define a character or not define there are characters already defined pick up a character which is one on entire h, but it is not true such a character always exist you just h is a subgroup you send h to 1 and then define you get these two will have many possibilities of choosing the character this character group size is same as this h q star. So, fix this now consider this quantity this is sum of 1 minus chi n whole times lambda n n less than equal to x. Now, let us think about this quantity for this sum for a moment and what we are interested in knowing is when is this sum all 0 or let us put it slightly differently suppose between 1 and n all elements of z q star are in h 1 n x intersection z q star is of that. Then what is this sum 0 right whenever n is in z q star chi n is 1. So, this is 0 whenever n is not in z q star and lambda n is 0 lambda n is non 0 only when n is the prime power if n is not in z q star it does not trivial g c d with q yes and we are saying that x is less than q q is smaller than anywhere else. Now, so this is 0 if this happens now all what we want to show is that in the range 1 to log n log q to the 4 there is an element which is not in z q not in the group h. So, if we can show that this sum is non 0 when x is log q to the 4 and then clearly it is less than q right that is the condition we want to ensure this will be there. So, if now to the 4 q. So, now let us see what this is equal to we already know what this is equal to what is this the first quantity the first sum equal to what first of all it is psi x and the second one is psi of x of chi that is by definition and now use g r h. If g r h is true then psi x equals x plus order square root x log square x and the g r h is true then y to psi x of chi. So, chi is non trivial for trivial character it is the same quantity x plus order square root x log square q x, but if chi is non trivial then this is simply order square root x log square q x which is x plus order square root x log square q x. Now, this would be 0 when this error term is equal to the principal term and it has opposite sign, but for certain see, but asymptotically the principal term is much bigger than the error term. So, at what point does it become bigger and stays bigger forever when x is less than square root x log square q x you can only if x is bigger than log to the 4 q x. So, if x is bigger than let us say c cube log to the 4 cube will that be sufficient log to the 4 cube or instead of c cube we can say it from 10 c square that is not very important and that is it. That means, if I look at the interval 1 to x where x is order log to the 4 cube I am guaranteed to find an element which is outside any sub proper subgroup of any proper subgroup of z cube star. So, on any side the sum should be 0 the elements for which n is the n as sum g c d with q and still it can be a prime power and attack the c d cube. How do you handle this? If n is a prime power and the entire prime this for this prime power to divide q that is the requirement. So, it will have a non free e to d and therefore, lambda n is not 1 chi what is chi n then chi n is 0 chi n by definition is 0. So, then this is not that is certainly the sum with v non 0. So, then I will have to count number of such occurrences. So, the sum will not be 0, but what one can see the sum is going to be very small. The x is only going up to log to the 4 cube. So, we are only looking at prime divisors of q which are very small and for each such prime divisor lambda n is log of that. So, if you sum it all up the sum of logarithm of all prime divisors of all small prime divisors of q that is going to be a very small quantity. It has to be a prime power. So, there are a small prime number and we are going up to only log to the 4 cube. So, that will let us say prime p divides which is prime p itself is very small it divides q. So, we will go up to a small power of p. The power of p we go up to is log of p and actually it does not matter what power of p we go up to because that is only log p we get counted. So, log p. So, p itself is less than equal to log to the 4 cube. So, log p is log log q and how many such primes should they be? They can be at most log q such primes because their product is going to be q. So, log q times log log q is the maximum value of this sum and then we go here and again do the same thing. So, 0 is not that important because this gap is going can be made fairly large. For an appropriately chosen constant here, this gap would be log to the 4 cube which is way beyond what is allowed. But that is a good point that we have to be careful. So, corollary of this is that you look at all elements of the q star less than equal to log order log to the 4 cube they must generate the entire group because if they do not then there is a subgroup and then that is important. And there are other many many other corollary like a quadratic non residue model of a primes. That you want to find given a prime p you want to find a quadratic non residue model of p which is used in factoring quadratic polynomial. If a if you have to factor polynomial like x square minus a over f p then the one convenient algorithm is that if you have available one quadratic non residue then using that non residue you can polynomial factor polynomial. Now, how do you get a quadratic non residue? You use this theorem start searching for 1, 2, 3, 4 and checking if a number is quadratic non residue. You know how to check a number if a number is quadratic non residue raise it to p minus 1 model of p and see you get plus 1 or minus 1 that is correct p minus 1 by 2 p minus 1 is always get here raise it p minus 1 by 2 and you get a plus 1 by 2. So, you keep checking now one of the first log to the 4 p number has to be quadratic non residue because set of quadratic residue form is proper subgroup of z p star. The other application is Miller's algorithm for final t testing and I am not going to describe the algorithm you go and look it up it is a deterministic algorithm. It is polynomial time and it claims that the g r which is true and the reason for the claim is that again tries to do the same thing it computes a subgroup of z p star or the number say z n star the n is the number if you want to check it is prime or not. So, it looks at the z n star then identify the subgroup proper subgroup of z n star and looking for an element outside this subgroup. If that element is available then the algorithm becomes polynomial time and this is the way to find that element. You just run this you know throw it big hammer of g r h at it you get a polynomial time. So, any questions? No, without g r h the best known is square root q. So, there are lots and lots of problem which will solution will become easy if g r h is true in all kinds of diverse fields. For example, finding say we know that between n and 2 n there exist a prime for every n, but what one would like to do is to shrink this interval and show the n and 2 n. Obviously, there must be a prime and there must be able to shrink this interval and say that between n and n plus something there must always be a prime. Now, if g r h is true then you can show that between n and n plus n to the 1 plus epsilon if I remember correctly there must be a prime for n and n plus n to the epsilon. So, the interval the gap is only n to the epsilon, but in fact g r h is not strong enough actually to fully satisfy us here because here what is believed is between n and n plus order log square n there must be a prime, but that does not follow from g r h, but it is believed to be correct. Now, of course I have said already much about the Riemann hypothesis, but one final thing I want to say is that it has some part from its fundamental importance to mathematics seems to have very curious connection with physics. There is a theory called quantum field theory in physics I do not know about that anything but what I have read it basically models the quantum phenomena and in that there is a particular governing matrix which is huge matrix and the Eigen values I say infinite matrix by the way and Eigen values of that matrix are precisely the 0's of the zeta function. So, that connection was discovered not too long ago and it was quite a remarkable insight that zeta function has something to say about the physical theory. So, anyway if you are interested feel free to read that up I still have time which is good because now I want to change track I have about 4 lectures and in this I want to like to do the following I will give you an example of zeta function of a different kind of zeta function that is an example is over elliptical and let me create a new template. So, in fact there is a full theory of this zeta function can be defined over many different objects in particular over all two dimensional curves and there is a corresponding Riemann hypothesis associated with it and that hypothesis I actually been proven and in fact I will prove the Riemann hypothesis for elliptic curves and if the Riemann hypothesis there also is the same that all the 0's non trivial 0's lie on the line real zeta equals 0. Now, elliptic curves are interesting for many other reasons as well and that is why I thought I will do this because once I prove this I will show you the connection between the zeta function for elliptic curve and not just zeta function elliptic curve actually any Dirichlet series Dirichlet series is the zeta function original expression of the kind sigma. So, we have that is for example, this one over. So, this such infinite sum is called a Dirichlet series and the reason it is called well because it was defined by Dirichlet but it is a infinite series and then there is another quantity called power series. This is a more familiar quantity corresponding to this the power series would be e of z equals sigma n greater than equal to 1 z to the n. So, z to the n is the familiar infinite power series. The Dirichlet series is a it is n to the z. So, it just flips the thing and takes it in denominator and in general there would be coefficients here like in case of L function we have chi option there. So, whatever coefficients are here we can stick those coefficients here. So, this Dirichlet series and power series you can create this duality between the two. And what has been shown is that if Dirichlet series has a certain structure which is that it is meromorphic over the entire complex plane whatever the function it can be extended to a meromorphic over the entire complex plane and it has satisfies a functional equation like the zeta function does and one more property I think the size of the coefficient should not be very large. Then the corresponding power series becomes what is called modular form and vice versa. So, again maybe I should not jump into this all the right away. Let us forget about this I will come back to this with me first focus on zeta function over elliptic curves. But eventually in the last lecture I want to connect all of this up to the Fermat's last theorem and how that was shown using the ideas from elliptic curves modular function zeta function and so on. So, now what are elliptic curves do all of you know the definition you know two of you. So, elliptic curves are cubic curves we all know completely the quadratic curves cycle parabola hyperbola ellipse these are the four types of quadratic curves. And the simplest next curve is elliptic curve typically given by equation y square equals A x y y square equals x cube plus A x plus B. And there is another condition we attach to this 4 A cube minus 27 B square should be non-zero that is essentially to ensure that the right hand side which is a degree 3 polynomial in x. So, it is a degree 3 polynomial in x it will have three roots. So, I can rewrite right hand side as x minus alpha x minus beta x minus gamma. So, this condition is equivalent to saying that there are the right hand side does not have repeated roots because if you have repeated roots in the reclip the curve is sort of degenerate. And we do not want that situation to occur that is all. So, it is a very reasonable condition to assume. So, how does an elliptic curve look like let us take some example suppose y square equals x x minus 1 x plus 1. So, I am just writing it in this form of course, this if you multiply this out it is in this form. So, it is x square minus 1 times. So, how does this curve look like. So, I am not over real I am not over complex. So, real how does this look like. So, clearly at x equal to 1 y is 0 x equal to 0 y is 0 and x equal to minus 1 y is 0 y is 0 at all these three points. For x between minus 1 and 0 what happens I should ask for the first one for x less than minus 1 what happens for x less than minus 1 this negative this is negative and this is negative the product is a negative number. And then y square equal to a negative number on over real there is no solution it does not exist solution. So, the curve does not exist to the left of minus 1 what about between 0 and 1 same it is the product is negative. So, again the curve does not exist between 0 and 1. So, do a little bit more thinking about it. So, we will find that the curve will look like something like this. So, this part is a closed curve and this part is an infinite and this is the general form of an elliptic curve. There is one piece which is closed curve and one piece which is an infinite curve. Now, these look like ellipse and parabola, but they are not these are cubic curve. So, this is not really an ellipse, but it is symmetric along x axis because it is y square. Now, elliptic curves are just probably the most remarkable curve in the whole of mathematics it is just unbelievable the kind of properties there. And they all arise because of a group structure which sits on this curve one can define. So, we look at let us see. So, let us say e of that is our real. So, we can say take any q is set of this notation e of q to denote the set of rational point lying on the elliptic curve given by when I say rational point I mean a point here with coordinates of r. So, this of course, can be generalized to say e of r. e of r is the entire curve all the points e of q is a subset of those points which are right. So, the remarkable property of elliptic curve is that e of q or e of r or even if you want e of c we can even read this equation over complex numbers and look at all the complex solutions. So, those are the complex points lying on the elliptic curve. So, these form a group under addition and the addition is not this is not usual addition. And actually this is also slightly cheating because e of q does not itself form a curve we have to throw in one more point to this. So, I should say pointed infinity along with all three any of the three basically look at the points here and then throw in one point which is pointed infinity. It is an abstract entity we have I will make it precise later on, but to sort of intuitively just imagine it being a point lying on the curve, but which is at infinity. So, now let me say what is the addition operation for the elliptic curve for that let us draw the curve which I described earlier. So, this is the elliptic curve and to add two points let us say this is point a this is point b you want to add n b to add n b the process is draw a line through n b this line is going to intersect the curve at the third point and this is the guaranteed why is it guaranteed because it is a cubic curve. So, it is guaranteed to intersect the equation of line is y equal to some c x m x plus c. So, plug that in for y into y square equal to x cube plus a x plus b we get a cubic in x it has three solutions one has to be either it has three solutions or it has one solution these are the two possibilities if it has three solutions. So, because it is this line is running through two points we cannot have one solution it has to have three solutions that is a guarantee third point must like this. So, that is the third point let us call it minus the reason I call minus t is we will be clear in a moment and now this is symmetric along x axis. So, there is a reflection of this point below which will call c and this point is the addition of a plus b equal to c. So, this process can be followed there are some degenerate cases one has to worry what if this point is here or let us say that the point is I want to add a with itself let us call it a prime. So, a prime plus a prime because it is a group addition it has to satisfy for all pairs. So, then again the geometric answer is simple draw tangent on a prime this tangent again is guaranteed to intersect the curve at a second point again reflected. So, these are three intersection scenario, but there is also one intersection scenario. For example, you take this point here a double prime and if you add a double prime to itself by just now what I defined you draw tangent at a double prime which is going to be a vertical line. So, it intersects the curve exactly one point which is a double prime there is no third point it intersects right and this is where that point and infinity comes to our help because this is a vertical line going to infinity which pretend that it is intersecting the curve at that infinite point. And this will only allow when the line is vertical the line joining two points is vertical then we say that the third point is because any line that joins that is vertical is not going to intersect the third point on the curve. So, that we pretend is intersecting at that point at infinity. Here a double prime itself is infinity. No a double prime is equal to minus a double prime. Yeah, but it is not infinity. So, a double prime minus infinity the point at infinity is unique. Point at infinity actually for this group acts at the identity element. So, a double prime is also equal to minus. A double prime is its own inverse. So, one can easily verify that infinity is the identity. Because if you take any point a and add infinity to it you should get the point a. So, how do you add a with infinity? Now, since infinity is associated with the vertical line. So, if you want to add a and infinity you draw a vertical line at a. Now, see what is the other point it intersects. It will intersect it at this point which is minus a and then the algorithm will tell us a reflected back reflecting it back you come back to a and that is the sum. So, a prime plus a plus infinity is a and same thing you can verify. Infinity plus infinity how do you visualize that? Infinity plus infinity must be infinity. Yeah, you just can just visualize it any way. It is like a point which never intersects a line here which is not intersecting it anywhere. So, that is you just pretend it is intersecting infinity at three times. No, intersect. Yeah, three times over as many times as you want. So, that these are all degenerate cases which you can just fix. And this geometric realization is just for our own understanding really. One can give a much more precise and rigorous algebraic definition. And there because some of these things seem to be after you which I am giving tell you but if you look at it algebraically it all does. So, this is the group that sits on an elliptical. In fact, there are lots of groups which sit on elliptical. I already identified three of them. The group over complex number, the group over real number and the group over rational. The group of rational point, the group of real point and group of complex. And one can go further and define elliptical over finite. Again, this is simply y square equal to x cube plus a x plus b a cube minus 27 b square is non-zero. What? Not necessarily, we can close the algebraic. So, we can get different groups on the field on the curve as we proceed towards closure. Just like as in case of real and complex. For real also these are defined. That is not algebraic. But I can close it to complex number and get another set of point. And those set of point also are common. Similarly, this is over field f of characteristic and p is I do not want p to be 2 or 3. Because p is 2 or 3 then this form of equation does not quite exist because you see that this discriminant gets messed up. The y square and x cube these are all get messed up. So, there are alternative forms of the curve which hold for characteristic 2 and 3. But for the sake of simplicity we will not consider. So, let us just concentrate on that. Note that f cube need not be the base field. Q must be the algebraic power of p. And now we can define. So, this is a caliptic curve over this field. Now, we can define talk about the set of point which are points on e in f cube. And not only f cube again as we already said we can consider because not all points may have not all values may have solution in f cube. That is you plug in some value of y. Now, you get a cubic in x. You get three roots. Not all those three values may be in f cube. They may lie in the higher field extension field. They will certainly lie in the algebraic closure of f cube. So, in general we can talk about extensions of f cube which are of the form f cube to the k. And points lying on the curve which are belong to f cube. Now, clearly e of f cube is the subset of e of f cube square. Let me just close by saying that the Riemann hypothesis which I am going to show. In the next class tomorrow. And I will prove it in the next class. It is not too difficult. We will have to assume some results about electrical, but that is not enough. We will be about the elliptic curves over finite field. And we will show nice the data function for the this such curves and the nice functional equation and lots of functions. Elliptic curves over rational are a different field. They are not very well understood or at least not as well understood as this. In fact, we do not even know there is a corresponding Riemann hypothesis there, but we do not know that Riemann hypothesis. We know something now. Thanks to this result of Andrew Weis and others, we know that the zeta function does have a functional equation, but that is about it.