Cfrgtkky

I want to show that for $D>0, Dequiv 0,1 pmod4$, $z in mathbb{C}$ with $ operatorname{Im}(z)>0$ and $k>1$ the sum

$$sum_{a,b,c in mathbb{Z} \ b^2-4ac=D}(az^2+bz+c)^{-k}$$ converges absolutely.

It is actually rather straightforward. If $a=0$, then $b$ can take at most 2 values, none of them $0$, and we are left with the absolutely convergent sum $sum_c |c+bi|^{-k}$. If $c=0$, we can run a similar argument. So it is enough to consider $a,b,cne 0$. In this case the roots $u,v$ of the corresponding quadratic polynomial $P(x)=ax^2+bx+c$ are real and lie at the distance $sqrt D$ from $-frac b{2a}$. Thus $|P(z)|=|a||z-u||z-v|$. This is never less than $|a||Im z|^2$ and for $|b|>(6sqrt D+4|Re z|)|a|$ is not less than $|a||frac b{4a}|^2approx frac{|b|^2}{|a|}$. Thus, $|P(z)|gtrsimmax(|a|,frac{|b|^2}{|a|})ge |b|$ and we only need to consider $sum_{a,b,c}|b|^{-k}$. However, for fixed $b$, the number of non-zero pairs $(a,c)$ with $0ne b^2-D=4ac$ is at most the number of divisors of $b^2-D$, which grows slower than any positive power of $|b|$.

This is an incomplete answer using the Theory of Quadratic Forms. The constants $C$ appearing in this solution may not be the same. We begin with

Definition

Let $ax^2+bxy+cy^2$ be a quadratic form with $a, b, cin mathbb{Z}$ and $b^2-4ac=D$. We say that a quadratic form $a_1x_1^2+b_1x_1y_1+c_1y_1^2$ with $a_1,b_1,c_1inmathbb{Z}$ and $b_1^2-4a_1c_1=D$ is equivalent to $ax^2+bxy+cy^2$ if there is $begin{bmatrix}p & q \ r & s end{bmatrix}inmathrm{SL}_2(mathbb{Z})$ such that
$$
a_1x_1^2+b_1x_1y_1+c_1y_1^2=a(px_1+qy_1)^2+b(px_1+qy_1)(rx_1+sy_1)+c(rx_1+sy_1)^2.
$$

This can be written again with matrices
$$
begin{bmatrix}a_1 & frac{b_1}2\ frac{b_1}2 & c_1 end{bmatrix} = begin{bmatrix}p & r \ q & s end{bmatrix} begin{bmatrix}a & frac{b}2\ frac{b}2 & c end{bmatrix} begin{bmatrix}p & q \ r & s end{bmatrix}
$$
and
$$
begin{bmatrix}x\yend{bmatrix}=begin{bmatrix}p & q \ r & s end{bmatrix}begin{bmatrix}x_1\y_1end{bmatrix}.
$$

The next one is the reducing step.

Lemma 1

Any integer quadratic form with discriminant $D$ is equivalent to $ax^2+bxy+cy^2$ with $|b|leq |a|leq |c|$.

The proof is a repeated application of the matrices in $mathrm{SL}_2(mathbb{Z})$.
$$
begin{bmatrix} 1 & 1 \ 0 & 1 end{bmatrix}, begin{bmatrix} 0 & -1 \ 1 & 0end{bmatrix}.
$$

Corollary 1
The number of equivalence classes of integer quadratic forms with discriminant $D$ is finite.

If $b^2-4ac=D$ and $|b|leq |a|leq |c|$, then $|D|geq 4|ac|-|b^2| geq 3|ac|$ gives only finitely many choices for $a$ and $c$, subsequently for $b$.

Then it suffices to prove the following for $A=begin{bmatrix}a& b/2\ b/2&cend{bmatrix}$ with $b^2-4ac=D$ and $|b|leq |a|leq |c|$, as the sum can be written as a finite sum of the following kind.

The sum is absolutely convergent for $k>1$:
$$sum_{gammain mathrm{SL}_2(mathbb{Z})} frac1{left(| gamma z - alpha | |gamma z - alpha'|right)^k|rz+s|^{2k}} (*)
$$
where $gamma z=frac{pz+q}{rz+s}$, $alpha=frac{-b-sqrt{D}}{2a}$, and $alpha'=frac{-b+sqrt{D}}{2a}$.

Note that after the reduction, we have $ac<0$ and $|alpha-alpha'|>1$.

Let $Delta_{gamma}$ be the area of a triangle with vertices at $alpha, alpha',gamma z$. Let $theta_{gamma}$ be the angle formed by the corner $alpha, gamma z, alpha'$. Then $Delta_{gamma}=frac12| gamma z - alpha | |gamma z - alpha'|sin theta_{gamma}=frac12 |alpha-alpha'|Im(gamma z)=frac12|alpha-alpha'| frac{Im(z)}{|rz+s|^2}$.

Thus, it suffices to prove the convergence of
$$
sum_{gammain mathrm{SL}_2(mathbb{Z})} (sin theta_{gamma})^k
$$

Consider $Gamma=mathrm{SL}_2(mathbb{Z})$ and $H=left{pmbegin{bmatrix} 1&2m_0m\0&1end{bmatrix}, minmathbb{Z}right}$. Fix a fundamental domain $F_{infty}$ for $H$ as $F_{infty}={z| u-m_0 le Re z < u+m_0}$ and $u=frac{alpha+alpha'}2$. Here we set the integer $m_0$ large enough that $u-m_0<min(alpha,alpha')<max(alpha,alpha')<u+m_0$.
For $gamma znotin F_{infty}$, $gamma z$ is sufficiently away from any of $alpha$, $alpha'$. Thus, we have
$$
sum_{(r,s)=1, begin{bmatrix} *&*\r&send{bmatrix} z notin F_{infty}}frac1{left(| gamma z - alpha | |gamma z - alpha'|right)^k|rz+s|^{2k}}ll sum_{(r,s)=1} frac1{|rz+s|^{2k}}
$$
due to $sum frac 1{n^k}$ is convergent. Moreover, the sum on the right side is
$$
ll sum_{(r,s)=1, rsneq 0} frac1{|rs|^k}
$$
which is also convergent. The implied constants depend on $k,alpha,alpha',m_0,z$.

Thus, we consider the following sum
$$
sum_{gammain Hbackslash Gamma} (sintheta_{gamma})^k=sum_{gammainGamma, gamma z in F_{infty}} (sin theta_{gamma})^k.
$$
We need a bound for the number of orbit points $gamma z$ in the following figure.