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.,

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

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

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

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

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

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

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

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

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

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

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

E-Mail:

kleitner@edv1.boku.ac.at

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