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The number of proper representations of any integer $n$ by all the positive definite binary quadratic forms of some fixed discriminant can be expressed as a similar divisor like sum involving Jacobi symbols using Dirichlet's Mass formula. Further since the arbitrary representations of $n$ by some quadratic form are proper representations of $n/d$ coordinate wise scaled by $\sqrt{d}$ for a square $d\mid n$. We can prove that if $r_2(n)=4(d_1(n)-d_3(n))$ is the number of integer pairs $(x,y)\in\mathbb{Z}^2$ which satisfy $x^2+y^2=n$ while if $s(n)=|\{d\mid n:\sqrt{d}\in\mathbb{N}\}|=\prod_{p\mid n}\left(1+\lfloor v_p(n)/2\rfloor\right)$ is the number of perfect squares dividing $n$ and $\omega(n)=|\{p\mid n:p\text{ is prime}\}|$ is the number of distinct primes dividing $n$ while for every negative integer $m$ defining:

$$w(m)=\begin{cases}6&\text{ if }d=-3\\4&\text{ if }d=-4\\2&\text{ otherwise}\end{cases}$$ $$\epsilon_m(n)=\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{m}{p}\right)\right)=\begin{cases}1&\text{ if }\exists x\in \mathbb{Z}:x^2\equiv m\bmod n\\0&\text{ if }\not\exists x\in \mathbb{Z}:x^2\equiv m\bmod n\end{cases}$$

Note if $f_1,f_2,\ldots f_{h(D)}$ are the reduced positive definite binary quadratic forms of discriminant $D$ where $h(d)$ is the class number of $D$ then for any odd integer $n\in\mathbb{N}$ coprime to $D$ we must have:

$$\small\sum_{k=1}^{h(D)}|\{(x,y)\in\mathbb{Z}^2:f_k(x,y)=n\land \gcd(x,y)=1\}|=w(D)\sum_{d\mid n}\left( \dfrac{D}{d}\right)=w(D)\epsilon_D(n)2^{\omega(n)}$$

$$\small\implies\sum_{k=1}^{h(D)}|\{(x,y)\in\mathbb{Z}^2:f_k(x,y)=n\}|=w(D)\sum_{d\mid n}\left( \dfrac{D}{d}\right)s(d)=w(D)\epsilon_D(n)\frac{r_2(n)}{4}$$

For instance if we look at those pdbqf's of discriminant $D=-28$ there is only one reduced form, namely $f(x,y)=x^2+7y^2$ since $\small h(-28)=1$ thus for any $\small n\in\mathbb{Z}$ with $\small (n,D)=1$ we must have by the previous formula that:

$$|\{(x,y)\in\mathbb{Z}^2:x^2+7y^2=n\land (x,y)=1\}|=2^{\omega(n)+1}\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{-7}{p}\right)\right)$$ $$|\{(x,y)\in\mathbb{Z}^2:x^2+7y^2=n\}|=\frac{r_2(n)}{2}\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{-7}{p}\right)\right)$$

For details on Dirichlet's formula see Chapter 4 of: http://www2.math.ou.edu/~kmartin/ntii/ntii.pdf

The number of proper representations of any integer $n$ by all the positive definite binary quadratic forms of some fixed discriminant can be expressed as a similar divisor like sum involving Jacobi symbols using Dirichlet's Mass formula. While since the arbitrary representations of any $n\in\mathbb{N}$ by a quadratic form are proper representations of $n/d$ coordinate wise scaled by $\sqrt{d}$ for a square $d\mid n$. We can prove that if $r_2(n)=4(d_1(n)-d_3(n))$ is the number of integer pairs $(x,y)\in\mathbb{Z}^2$ which satisfy $x^2+y^2=n$ while if $s(n)=|\{d\mid n:\sqrt{d}\in\mathbb{N}\}|=\prod_{p\mid n}\left(1+\lfloor v_p(n)/2\rfloor\right)$ is the number of perfect squares dividing $n$ and $\omega(n)=|\{p\mid n:p\text{ is prime}\}|$ is the number of distinct primes dividing $n$ then for every integer $m$ coprime to $n$ we define:

$$w(m)=\begin{cases}6&\text{ if }d=-3\\4&\text{ if }d=-4\\2&\text{ otherwise}\end{cases}$$ $$\epsilon_m(n)=\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{m}{p}\right)\right)=\begin{cases}1&\text{ if }\exists x\in \mathbb{Z}:x^2\equiv m\bmod n\\0&\text{ if }\not\exists x\in \mathbb{Z}:x^2\equiv m\bmod n\end{cases}$$

Note if $f_1,f_2,\ldots f_{h(D)}$ are the reduced positive definite binary quadratic forms of discriminant $D$ where $h(d)$ is the class number of $D$ then for any odd integer $n\in\mathbb{N}$ coprime to $D$ we must have:

$$\small\sum_{k=1}^{h(D)}|\{(x,y)\in\mathbb{Z}^2:f_k(x,y)=n\land \gcd(x,y)=1\}|=w(D)\sum_{d\mid n}\left( \dfrac{D}{d}\right)=w(D)\epsilon_D(n)2^{\omega(n)}$$

$$\small\implies\sum_{k=1}^{h(D)}|\{(x,y)\in\mathbb{Z}^2:f_k(x,y)=n\}|=w(D)\sum_{d\mid n}\left( \dfrac{D}{d}\right)s(d)=w(D)\epsilon_D(n)\frac{r_2(n)}{4}$$

For instance if $D=-28$ then $f(x,y)=x^2+7y^2$ is the only reduced form of discriminant $D$ because $h(-28)=1$ thus for every odd integer $n>0$ not divisible by $7$, we have that:

$$|\{(x,y)\in\mathbb{Z}^2:x^2+7y^2=n\land (x,y)=1\}|=2^{\omega(n)+1}\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{-7}{p}\right)\right)$$ $$|\{(x,y)\in\mathbb{Z}^2:x^2+7y^2=n\}|=\frac{r_2(n)}{2}\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{-7}{p}\right)\right)$$

For details on Dirichlet's formula see Chapter 4 of: http://www2.math.ou.edu/~kmartin/ntii/ntii.pdf

The number of proper representations of any integer $n$ by all the positive definite binary quadratic forms of some fixed discriminant can be expressed as a similar divisor like sum involving Jacobi symbols using Dirichlet's Mass formula. Further since the arbitrary representations of $n$ by some quadratic form are proper representations of $n/d$ coordinate wise scaled by $\sqrt{d}$ for a square $d\mid n$. We can prove that if $r_2(n)=4(d_1(n)-d_3(n))$ is the number of integer pairs $(x,y)\in\mathbb{Z}^2$ which satisfy $x^2+y^2=n$ while if $s(n)=|\{d\mid n:\sqrt{d}\in\mathbb{N}\}|=\prod_{p\mid n}\left(1+\lfloor v_p(n)/2\rfloor\right)$ is the number of perfect squares dividing $n$ and $\omega(n)=|\{p\mid n:p\text{ is prime}\}|$ is the number of distinct primes dividing $n$ while for every $m\in\mathbb{N}$ defining:

$$w(m)=\begin{cases}6&\text{ if }d=-3\\4&\text{ if }d=-4\\2&\text{ otherwise}\end{cases}$$

Note if $F=\small\{f_1,\ldots f_{h(D)}\}$ are the reduced positive definite binary quadratic forms of discriminant $D$ where $h(d)$ is the class number of $D$ then for any odd integer $n\in\mathbb{N}$ coprime to $D$ we must have:

$$\small\sum_{f\in F}|\{(x,y)\in\mathbb{Z}^2:f(x,y)=n\land \gcd(x,y)=1\}|=w(D)\sum_{d\mid n}\left( \dfrac{D}{d}\right)=w(D)2^{\omega(n)}\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{D}{p}\right)\right)$$

$$\small\implies\sum_{f\in F}|\{(x,y)\in\mathbb{Z}^2:f(x,y)=n\}|=w(D)\sum_{d\mid n}\left( \dfrac{D}{d}\right)s(d)=\frac{w(D)r_2(n)}{4}\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{D}{p}\right)\right)$$

For instance if we look at those pdbqf's of discriminant $D=-28$ there is only one reduced form, namely $f(x,y)=x^2+7y^2$ since $\small h(-28)=1$ thus for any $\small n\in\mathbb{Z}$ with $\small (n,D)=1$ we must have by the previous formula that:

$$|\{(x,y)\in\mathbb{Z}^2:x^2+7y^2=n\land (x,y)=1\}|=2^{\omega(n)+1}\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{-7}{p}\right)\right)$$ $$|\{(x,y)\in\mathbb{Z}^2:x^2+7y^2=n\}|=\frac{r_2(n)}{2}\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{-7}{p}\right)\right)$$

For details on Dirichlet's formula see Chapter 4 of: http://www2.math.ou.edu/~kmartin/ntii/ntii.pdf

The number of proper representations of any integer $n$ by all the positive definite binary quadratic forms of some fixed discriminant can be expressed as a similar divisor like sum involving Jacobi symbols using Dirichlet's Mass formula. Further since the arbitrary representations of $n$ by some quadratic form are proper representations of $n/d$ coordinate wise scaled by $\sqrt{d}$ for a square $d\mid n$. We can prove that if $r_2(n)=4(d_1(n)-d_3(n))$ is the number of integer pairs $(x,y)\in\mathbb{Z}^2$ which satisfy $x^2+y^2=n$ while if $s(n)=|\{d\mid n:\sqrt{d}\in\mathbb{N}\}|=\prod_{p\mid n}\left(1+\lfloor v_p(n)/2\rfloor\right)$ is the number of perfect squares dividing $n$ and $\omega(n)=|\{p\mid n:p\text{ is prime}\}|$ is the number of distinct primes dividing $n$ while for every negative integer $m$ defining:

$$w(m)=\begin{cases}6&\text{ if }d=-3\\4&\text{ if }d=-4\\2&\text{ otherwise}\end{cases}$$ $$\epsilon_m(n)=\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{m}{p}\right)\right)=\begin{cases}1&\text{ if }\exists x\in \mathbb{Z}:x^2\equiv m\bmod n\\0&\text{ if }\not\exists x\in \mathbb{Z}:x^2\equiv m\bmod n\end{cases}$$

Note if $f_1,f_2,\ldots f_{h(D)}$ are the reduced positive definite binary quadratic forms of discriminant $D$ where $h(d)$ is the class number of $D$ then for any odd integer $n\in\mathbb{N}$ coprime to $D$ we must have:

$$\small\sum_{k=1}^{h(D)}|\{(x,y)\in\mathbb{Z}^2:f_k(x,y)=n\land \gcd(x,y)=1\}|=w(D)\sum_{d\mid n}\left( \dfrac{D}{d}\right)=w(D)\epsilon_D(n)2^{\omega(n)}$$

$$\small\implies\sum_{k=1}^{h(D)}|\{(x,y)\in\mathbb{Z}^2:f_k(x,y)=n\}|=w(D)\sum_{d\mid n}\left( \dfrac{D}{d}\right)s(d)=w(D)\epsilon_D(n)\frac{r_2(n)}{4}$$

For instance if we look at those pdbqf's of discriminant $D=-28$ there is only one reduced form, namely $f(x,y)=x^2+7y^2$ since $\small h(-28)=1$ thus for any $\small n\in\mathbb{Z}$ with $\small (n,D)=1$ we must have by the previous formula that:

$$|\{(x,y)\in\mathbb{Z}^2:x^2+7y^2=n\land (x,y)=1\}|=2^{\omega(n)+1}\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{-7}{p}\right)\right)$$ $$|\{(x,y)\in\mathbb{Z}^2:x^2+7y^2=n\}|=\frac{r_2(n)}{2}\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{-7}{p}\right)\right)$$

For details on Dirichlet's formula see Chapter 4 of: http://www2.math.ou.edu/~kmartin/ntii/ntii.pdf

The number of proper representations of any integer $n$ by all the positive definite binary quadratic forms of some fixed discriminant can be expressed as a similar divisor like sum involving Jacobi symbols using Dirichlet's Mass formula. Further since the arbitrary representations of $n$ by some quadratic form are proper representations of $n/d$ coordinate wise scaled by $\sqrt{d}$ for a square $d\mid n$. We can prove that if $r_2(n)=4(d_1(n)-d_3(n))$ is the number of integer pairs $(x,y)\in\mathbb{Z}^2$ which satisfy $x^2+y^2=n$ while if $s(n)=|\{d\mid n:\sqrt{d}\in\mathbb{N}\}|=\prod_{p\mid n}\left(1+\lfloor v_p(n)/2\rfloor\right)$ is the number of perfect squares dividing $n$ and $\omega(n)=|\{p\mid n:p\text{ is prime}\}|$ is the number of distinct primes dividing $n$ while for every $m\in\mathbb{N}$ defining:

$$w(m)=\begin{cases}6&\text{ if }d=-3\\4&\text{ if }d=-4\\2&\text{ otherwise}\end{cases}$$

We see if $f_1,f_2,\ldots f_{h(D)}$ are the reduced positive definite binary quadratic forms of discriminant $D$ where $h(d)$ is the class number of $D$ then for any odd integer $n\in\mathbb{N}$ coprime to $D$ we must have:

$$\small\sum_{k=1}^{h(D)}|\{(x,y)\in\mathbb{Z}^2:f_k(x,y)=n\land (x,y)=1\}|=w(D)\sum_{d\mid n}\left( \dfrac{D}{d}\right)=w(D)2^{\omega(n)}\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{D}{p}\right)\right)$$

$$\small\implies\sum_{k=1}^{h(D)}|\{(x,y)\in\mathbb{Z}^2:f_k(x,y)=n\}|=w(D)\sum_{d\mid n}\left( \dfrac{D}{d}\right)s(d)=\frac{w(D)r_2(n)}{4}\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{D}{p}\right)\right)$$

For instance if we look at those pdbqf's of discriminant $D=-28$ there is only one reduced form, namely $f(x,y)=x^2+7y^2$ since $\small h(-28)=1$ thus for any $\small n\in\mathbb{Z}$ with $\small (n,D)=1$ we must have by the previous formula that:

$$|\{(x,y)\in\mathbb{Z}^2:x^2+7y^2=n\land (x,y)=1\}|=2^{\omega(n)+1}\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{-7}{p}\right)\right)$$ $$|\{(x,y)\in\mathbb{Z}^2:x^2+7y^2=n\}|=\frac{r_2(n)}{2}\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{-7}{p}\right)\right)$$

For details on Dirichlet's formula see Chapter 4 of: http://www2.math.ou.edu/~kmartin/ntii/ntii.pdf

The number of proper representations of any integer $n$ by all the positive definite binary quadratic forms of some fixed discriminant can be expressed as a similar divisor like sum involving Jacobi symbols using Dirichlet's Mass formula. Further since the arbitrary representations of $n$ by some quadratic form are proper representations of $n/d$ coordinate wise scaled by $\sqrt{d}$ for a square $d\mid n$. We can prove that if $r_2(n)=4(d_1(n)-d_3(n))$ is the number of integer pairs $(x,y)\in\mathbb{Z}^2$ which satisfy $x^2+y^2=n$ while if $s(n)=|\{d\mid n:\sqrt{d}\in\mathbb{N}\}|=\prod_{p\mid n}\left(1+\lfloor v_p(n)/2\rfloor\right)$ is the number of perfect squares dividing $n$ and $\omega(n)=|\{p\mid n:p\text{ is prime}\}|$ is the number of distinct primes dividing $n$ while for every $m\in\mathbb{N}$ defining:

$$w(m)=\begin{cases}6&\text{ if }d=-3\\4&\text{ if }d=-4\\2&\text{ otherwise}\end{cases}$$

Note if $F=\small\{f_1,\ldots f_{h(D)}\}$ are the reduced positive definite binary quadratic forms of discriminant $D$ where $h(d)$ is the class number of $D$ then for any odd integer $n\in\mathbb{N}$ coprime to $D$ we must have:

$$\small\sum_{f\in F}|\{(x,y)\in\mathbb{Z}^2:f(x,y)=n\land \gcd(x,y)=1\}|=w(D)\sum_{d\mid n}\left( \dfrac{D}{d}\right)=w(D)2^{\omega(n)}\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{D}{p}\right)\right)$$

$$\small\implies\sum_{f\in F}|\{(x,y)\in\mathbb{Z}^2:f(x,y)=n\}|=w(D)\sum_{d\mid n}\left( \dfrac{D}{d}\right)s(d)=\frac{w(D)r_2(n)}{4}\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{D}{p}\right)\right)$$

For instance if we look at those pdbqf's of discriminant $D=-28$ there is only one reduced form, namely $f(x,y)=x^2+7y^2$ since $\small h(-28)=1$ thus for any $\small n\in\mathbb{Z}$ with $\small (n,D)=1$ we must have by the previous formula that:

$$|\{(x,y)\in\mathbb{Z}^2:x^2+7y^2=n\land (x,y)=1\}|=2^{\omega(n)+1}\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{-7}{p}\right)\right)$$ $$|\{(x,y)\in\mathbb{Z}^2:x^2+7y^2=n\}|=\frac{r_2(n)}{2}\prod_{p\mid n}\frac{1}{2}\left(1+\left( \dfrac{-7}{p}\right)\right)$$

For details on Dirichlet's formula see Chapter 4 of: http://www2.math.ou.edu/~kmartin/ntii/ntii.pdf