## 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 , we consider the set of positive definite, binary quadratic forms with integral coefficients of discriminant , i.e.,

Two forms , are called equivalent, if and only if there is a matrix , such that

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

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

culminating in Quite recently, Chamizo and Iwaniec improved this classical upper bound to

where . The main object of the present paper is to prove a two-sided Omega estimate for the error term .

For real we have

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

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