Edit: I asked my question on mathoverflow(On Siegel mass formula), and the answers are satisfying to me.
I am interested deeply in the following problem:
Let $f$ be a (fixed) positive definite quadratic form; and let $n$ be any arbitrary natural number; then find a closed formula for number of solutions to the equation $f=n$.
For special case $f_1(x,y)=x^2+y^2$, here gives a closed formula for number of solutions.
Also you can find another formulas
for the special cases $f_5(x,y)=x^2+5y^2$ and $f_7(x,y)=x^2+7y^2$ there.You can finde a close formula here for $f_2(x,y)=x^2+2y^2$, here.
You can finde a close formula here for $f(x,y,z,w)=x^2+y^2+z^2+w^2$, here.
By a more Intelligently search through the web; you can find similar formulas for only finite limited number of positive definite quadratic forms.
[I think there exists such an explicit formula
at most for $10000$ quadratic forms. Am I right?]
As I have mentioned (I am not sure of it!) only for finite number of quadratic forms we have such a explicit, closed, nice formula; and this way goes in dead-end for arbitrary quadratic forms.
So Dirichlet tries to find the (weighted) sum of such representations by binary quadratic forms of the same discriminat.
That formula works very nice for our purpose if the genera contains exactly one form. In the dirichlet formula each binary quadratic forms apears by weight one in the (weighted) sum.
More precisely let $f_1, f_2, ..., f_h=f_{h(D)}$ be a complete set of representatives for binary quadratic forms of discriminant $D < 0$;
then for every $n \in \mathbb{N}$, with $\gcd(n,D)=1$ we have:
$$ \sum_{i=1}^{h(D)} N(f_i,n) = \omega (D) \sum_{d \mid n} \left( \dfrac{D}{d}\right) ; $$
where $\omega (-3) =6$ and $\omega (-4) =4$ and for every other (possible) value of $D<0$ we have $\omega (D) =2$. Also by $N(f,n)$; we meant number of integral representations of $n$ by $f$; i.e. :
$$ N(f,n) := N\big(f(x,y),n\big) = \# \{(x,y) \in \mathbb{Z}^2 : f(x,y)=n \} . $$
I have hered that there is a generalization of dirichlet's theorem for ;quadratic forms, having more variables; due to Siegel.
I have searched through the web;
but I have found only this link : Smith–Minkowski–Siegel mass formula ;
also I confess that I can't understand whole of this wiki-article.
Could anyone introduce me a reference in english; for Siegel mass formula ?