[SOUND] [MUSIC] Now we consider the partition function, Of the same problem, theta a tau zeta. As before, l is an even integral positive definite, like this. Tau this is an element in the upper half plane. But this calligraphic zeta is a vector in the complexification of the lattice L. This is a complex quadratic space of rank n. By definition, the partition function theta L tau zeta, is defined as the falling sum of all vectors in the lattice L e to the power pi(v,v) tau + 2(v,z). This is a new function now. Certainly like in the first generating function, we have this term, v to the square tau. But now we have an additional term. In principle, we cannot simplify these Fourier expansion. So partition function is defined by its Fourier expansion and all Fourier coefficients are 1 here. So we cannot construct, we cannot simplify the Fourier expansions. So let's analyze once more the definition of partition function. So in the partition function, we, in that generating function of the lattice L, the generating function is also defined the similar way. But we can combine different terms in this sum with the same scalar square of the vector v. Now, in the partition function we cannot do this. [SOUND] So it means that we can use this function to analyze the distribution. Distribution of v with fixed, Norm. So in this function, what is exactly an example of so-called Jacobi Modular Form. So what property has this function? First of all dot theta dot z of the lattice L is holomorphic, On the direct product of H1 and the complex quadratic space, L C of dimension m. And as holomorphic function, this function also has some hidden symmetries. Or it's better to say it satisfies two functional equations. The first equation is similar to the modular equation of the generating function, but it has the following form. We analyze the value of this function 1/2, but we have modified the vector z also by 2. And then this = tau to the power n over 2, like in the case of the function, times e to the power pi the scalar square of the complex vector z over 2, theta tau z. Certainly we have this property for an even unimodular lattice. So where's U? Since it's unilattice, it's equal to the original lattice. So this is a module equation of this theta function but this theta function, s theta function, if you worked a little bit with them, also has another function equation it satisfies. The following equation for the variable z. For lambda and mu in the lattice L, we can calculate this way. So it means that we can consider theta function of the lattice L as needed double periodic function with respect to that. So it means that the period here. I'm sorry, we have another lattice in this case certainly, this is tau L + L. So the functional equation is the following, e to the power- pi lambda, to the square 2 + 2 lambda z. Those are just two frontal equations of the partition function of this lattice. I call this function the partition function because we can control the distribution of our vector with fixed norm. I can formulate the first exercise, To prove this identity. So this is elementary question. But to prove this identity we have to use so-called poisson summation formula. Now I would like to understand the relation between the partition function of the lattice L and the usual theta function, or the arithmetic distribution function. [SOUND] So for this, we have to consider the pull backs, Of the petition function theta L tau z. The first pullback is quite natural. Let's put, z = 0. This is a 0 vector in the complex quadratic space. Then theta L tau 0 is equal, To the generating function or to the usual theta function, theta L tau. So from this point of view, the partition function theta is a generalization of the generating function for the number of the representation of natural numbers by the quadratic lattice L. But we can consider another pullback. Let u be a vector of L with square 2m, which is positive. Then we can define the following pullback of the partition function. So it means that, we put z equal to U times z where is complex number. Then, this function is equal to the following sum of all vectors in v. And here we get the scalar product of v and u times z. I can denote this function by theta L u(tau, z). And this is the so-called Jacobi Theta series, Of Eichler, And Zagier. Now, we considered this function in some details. [MUSIC]
