15. ÖMG-Kongress
Jahrestagung der Deutschen Mathematikervereinigung

16. bis 22. September 2001 in Wien

Sektion 2 - Zahlentheorie
Donnerstag, 20. September 2001, 15.30, Hörsaal 41


On the class number of binary quadratic forms

Manfred Kühleitner, Universität für Bodenkultur,Wien

For each positive integer $ n$, we consider the set $ {\cal Q}_n$ of positive definite, binary quadratic forms with integral coefficients of discriminant $ -n$, i.e.,

$\displaystyle {\cal Q}_n=\{aX^2+bXY+cY^2:\, b^2-4ac=-n\ \ \hbox{\rm and}\ \ a>0\}\, .$

Two forms $ AX^2+BXY+CY^2$, $ aX^2+bXY+cY^2$ are called equivalent, if and only if there is a matrix $ S\in {SL}_2(\mathbb{Z})$, such that

$\displaystyle AX^2+BXY+CY^2=(X,Y)S^t \left(\begin{array}{cc}a &b/2\\  b/2 & c\end{array}\right) S
\left(\begin{array}{c}X\\  Y\end{array}\right) \, .$

For a given discriminant $ -n$, the number $ N(n)$ of equivalence classes is finite. To study the average order of this arithmetic function, we consider the Dirichlet summatory function

$\displaystyle A(t)=\sum_{n\leq t}N(n)\, ,$

where $ t$ is a large real variable. In his masterwork Disquisitiones Arithmeticae, C.F. Gauß stated an approximate formula for $ A(t)$. In this century I. M. Vinogradov proved several upper bounds for the error term

$\displaystyle E(t):=A(t)-{\pi\over 18}t^{3/2}+{1\over 4}t$

culminating in $ E(t)\ll t^{2/3+\epsilon}\, .$ Quite recently, Chamizo and Iwaniec improved this classical upper bound to

$\displaystyle E(t)\ll t^{21/32+\epsilon}\, ,$

where $ 21/32=0.65625$. The main object of the present paper is to prove a two-sided Omega estimate for the error term $ E(t)$.

For real $ t\to\infty$ we have

$\displaystyle E(t)=\Omega_\pm\left(\sqrt{t\log t}\right)\, . $

E-Mail: kleitner@edv1.boku.ac.at

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