[SOUND] So that the Dedekind function is given in this formula by the infinite product. Such will have a good converges, this is product, an absolute and normal converges. So, now we can calculate. D of this product as usual. So, now let's try to write down the explicative formula, because an action, because the algorithmic. How the Dedekind theta function and we can use the usual rules of duration. What we get first of all, the first term, the first factor. Here is Q to the power 1 over 24 and we get here 1 over 24 to the [INAUDIBLE], then + This sum, N great to 1, I have to analyze this term. What we get? I get minus N Q to the power n and over 1 -q to the power n. This is the sum which we get, after the logarithm derivative. Now, I would like to written down, in more form. So, this is 1 over 24- sum n great =one, n q to the power n, and here you have the sum q to the power m, m great =0. So I will write the vector 1 over 1-q to the power n. I am sorry, here, we will certainly have to add the factor n. This is equal 1 over 24, I would like to write the double sum here. N and m both are greater to 1 n times q to the power m n. Why we have this? Because if we take the product. We get q to the power n (m+1), m is non negative. It means that (m+1) is great equal to 1. Then, they change the variable and they have this formula. Now, I would like to change the index of summation. So, here I don't know this product by capital n. Maybe let us use another color. Then, this is equal to 1 over 24- the sum over all capital n greater or equal to 1. Then, we have the sum of all devisers D of N. D q to the power of capital N all in this stand of notation, I'll get 1 over 24- the sum small and greater =1, sigma 1 of m q to the power n. Where sigma K in m is the definition. The sum over k's power was a positive divisor of n. So we have a function. Which looks like the Eisenstein series. So, I remind you that we have used in our Eisenstein series weight four and six. For example,Eisenstein series of weight of four following Fourier expansion. 1 + 240 is the sum, sigma 3 n, q to the power [INAUDIBLE] greater than or equal to 1. This is a modular form, of [INAUDIBLE] 4 with respect to the full modular group. A similar function with for weight six, eight, and so on. The Eisenstein series, so this function looks very similar to the Eisenstein series. The only difference, then, we have another normalization. So before sigma, before the sum, we have the constant 1 here. But here, the constant is 240. So, if we normalize this function, here we get 24. Let me use another notation for this function. By definition. G2 in tau. This is- the logarithmic derivative. Of the Dedekind theta function. So we would like to normalize it by constant 1 before sigma. -1 / 24 + the sum sigma 1 and (n) q to the power n. N greater equal to 1. Now, we can analyze, once more,the differential operator. Because the main operator which we defined was operator DK, and we have these formula. This formula quasi operator. Let me rewrite this formula once more. So our modular operator DK. F, this is D. F, now I have to change the sign, because we put here -. So we have +2k multiplication by z is quasi modular Eisenstein series. G2, why it's quasi modular, we'll see later. Through this is a modular form, a weak modular form. We have checked that this is weak modular form, but now I can write down its Fourier expansion. So what do we get for Fourier expansion of this function? First of all, we have the sum n times a(n) q to the power n, n greater or equal to 1. So cancel here the coefficient a0, the constant term + 2k Fourier expansion we have here- 1/24. Let me write it once more to remember it. And then Fourier expansion of the function F. So we see that this function is holomorphic at infinity. Because we have no negative powers of q. So we prove the following theorum. Dk. Is it operator from the space of modular form of weight k. Into the space of modular form of weight k + 2. Moreover, if f is a cusp point. Then, the image is a cusp form. Because q to the power 0 we could have only here. N = 0, we have here q to the power 0. But if A0 is = 0, but have no constant term in the Fourier expansion of the of F. So we construct it, the modular operator, the modular differential operator. Let us considered some examples. Let's calculate the image of the Eisenstein series of weight 4 This is D (E4) + 2k 8 G2 times E4. And we know that this is a modular form of weight 6, but this space is generated by the Eisenstein series of weight 6. So to calculate the image, we have to calculate only the constant term. And the constant term in the first here after the action of D. We have no constant term. So, the constant term, we can calculate using this product. And the constant term of this function is equals to 8 times -1 over 24 + q subset. So, the Eisenstein series is 6 has the constant term 1. So, this is =- E6,over 3. Using the same idea so exercise. You can calculate, the action of D6 on E6, D8. On E 8 D10 on E10. So here, you have to be very careful because this result, this function let me show you the problem. This function is a modular form of weight 12 and this space is generated by two functions. By the Eisenstein series of weight 12 and the Ramanujan delta function. So you have to calculate two Fourier coefficient. But, what do you have with Ramanujan delta function? Certainly, you can calculate it by definition, D12 delta. Because D is a logarithmic derivative G2 is algorithmic derivative of eta, but without explicit calculations. So certainly, you can use, first of all, the fact the delta function is in 24th power of eta. Or, you can use the fact that this function is a modular form of weight 14. Moreover, delta is a cusp form. Then, according to our theorem. This function is a cusp form of weight 14, with respect to full modular group. But this space [INAUDIBLE]. So you see there, D delta plus 24 G2 times delta is =0 identically. But certainly, you can check this identity directly by the definition. So we have very nice separator and now we have the following problem. What is the Eisenstein series G2, let us analyze the modular properties of this action. [SOUND] [MUSIC]
