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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 ?

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    $\begingroup$ Given $f_1$ I'd say you want to find some quadratic forms $f_2,\ldots,f_l$ such that $h(z) = \sum_{j=1}^l \Theta_{f_j}(z) =\sum_{j=1}^l \sum_{m \in \mathbb{Z}^n} e^{2i \pi z f_j(m,m)}$ is a Hecke eigenform so its coefficients are multiplicative, and the same for $h_\chi(z) = \sum_{j=1}^l \chi(j)\Theta_{f_j}(z)$ for some characters $\chi$. $\endgroup$ – reuns Sep 29 '17 at 17:01
  • $\begingroup$ @reuns ; My dear reuns, I have added something to the main text; I look for something as follows, Is Sigel mass formula similar to the following? Let $f_1,f_2,...,f_h$ be a complete set of representatives for binary quadratic forms of discriminant $D<0$; then $$\sum_{i=1}^{h(D)}N(f_i,n)=\omega (D)\sum_{d \mid n}\left( \dfrac{D}{d}\right);$$ 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 $f$. $\endgroup$ – Davood Khajehpour Sep 29 '17 at 18:03
  • $\begingroup$ @reuns ; could you explain your comment in more detail? Thanks. $\endgroup$ – Davood Khajehpour Oct 7 '17 at 17:25
  • $\begingroup$ @Will Jagy ; Could you help me on this question, or could you introduce me an english reference? $\endgroup$ – Davood Khajehpour Oct 9 '17 at 12:30
  • $\begingroup$ What you wrote in your comment is in the case $f_1$ is the quadratic form of a (fractional) ideal of $\mathbb{Q}(\sqrt{-D})$ ? (for an imaginary quadratic ideal $I = u_1\mathbb{Z}+u_2 \mathbb{Z}$ its quadratic form is $Q(n,n) = |u_1n_1+u_2n_2|^2$,$Q(n,m) = \frac{Q(n+m,n+m)-Q(n,n)-Q(m,m)}{2}$) $\endgroup$ – reuns Oct 9 '17 at 23:03

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