[SOUND] This type of function, data lu, was considered. In the book of Ziegler and now I would like to analyze this function as generating function. The generating function for what problem? First of all let's consider our first function. Theta in [INAUDIBLE]. Then all Fourier coefficients of this function are equal to r. And 2 n q to the power n. Where n is greater than or equal to 0. And where r if the number of representation of 2 n by the quadratic lead is A. What [INAUDIBLE] expansion we get for the height is right here, theta function. [NOISE]. If we analyze. The definition of this function we'll have to fix. Zen square over the N, this scale of product of v and u. [INAUDIBLE] in integer. So and we can collect the e function with the same, 2 n and l. We get the following Fourier expansion for the function. The sum is taken over all n. And l is that r l u n l q to the power n r to the power l, q in both cases. This is e to the power 2 pi i tau. And r = e to the power 2 pi i z. So this is the second, the exponent of the second variable that of our theta function. The petition function, the [INAUDIBLE] are this is the arithmetic function of the following time. This is a number. All vectors in L such as that square is equal to 2n and the scalar product with u is equal to l. So we analyze a distribution of the vector with fixed known and a fixed scalar product with vector u fix. So what function do we get here? Why this function, for example is interesting, for this we have to analyze one very, very classical, and very, very important. Ladies, but for these I will like maybe we'll align this later since i'm [INAUDIBLE] later, but now I would like only to fix it. So, the duration for E8, this is an even integral like this. Integral like this of determinant 1. Let us assume that E8 is equal to L8. So we can see the corresponding the series for L equal to E8. And now, let's assume that u is the root. Of this latest root means a vector of square two. Then maybe you know, maybe not, but we discuss as I told you, we analyze all these lattices and some details later. The to you in E8 this is so called latest. So let us analyze the function r for the. E8 and for the root u. For n equal to zero. For n equal to zero. This is a number of all vectors in EA with fixed norm 2 n and was the [INAUDIBLE] product 0 was. This is 0 vectors of the known 2n [INAUDIBLE] vectors belong to e7. Therefore we get the number for presentation. Of 2n by the latest E7 so studying the Jukubitus series of this type we can analyze this function is the number for presentation by E7, which is much more complicated than the number for presentation by E8. In our course for example, we get an explicit formula possess arithmetic function. The generated competition function. Our two meditation for the series of Jacobi forms but now I would like to give you one more motivation. I would like to analyze elliptization of the Ramanujan delta function. But to consider this elliptization I need the definition of modular form. Classical modular form in rate, so now we consider the modular form with respect with the modular group SL2C. Definition. A modular form, Of weight, k. K is an integer, with respect, there's a modular group s into the is a holomorphic function. F on the upper half plane which satisfies, The following modular equation. For any a, b, c, d methods in answer to that, and which has the following functional equation, yeah. So it's equal to the sum A and E to the power of two-pi-i and that. The summation is taken for all non-negative N. This is the very classical definition. And first of all, why we need this additional, conditional Fourier expansion? FIrst of all, according to the first equation for the translation t, f is periodic and we consider. And this expansion tells us that the function F of tau is holomorphic at e infinity or in another word than the function. Field q where q is equal to e plus the power of 2 p i. So. And q equal to zero. [MUSIC]