15. ÖMG-Kongress
Jahrestagung der Deutschen Mathematikervereinigung

16. bis 22. September 2001 in Wien

Sektion 2 - Zahlentheorie
Donnerstag, 20. September 2001, 16.30, Hörsaal 41


Zur Verteilung der Quadrate ganzer hyperkomplexer Zahlen

Gerald Kuba, Universität für Bodenkultur, Wien


Let $ \,{\cal A}\,$ be a real algebra of order $ \,s\geq 3\,$ where the basal units $ \,u_i\,(0\leq i<s)\,$ satisfy the primary Hamilton relations $ \; u_0u_i=u_iu_0=u_i\;(0\leq i<s),\;
\,u_iu_j=-u_ju_i\;(0<i<j<s),\;$ and $ \,u_i^2=-u_0\;(0<i<s)\,.\,$ The most important examples of such algebras are of course the division algebra $ \,\mathbb{H}\,$ of Hamilton's quaternions in dimension $ \,s=4\,$ and the division algebra $ \,\mathbb{O}\,$ of Cayley's octaves in dimension $ \,s=8\,$. Further let $ \,\Gamma\subset{\cal A}\,$ be any integral domain, for instance the Lipschitz ring or the Hurwitz ring of integral quaternions in the case $ \,{\cal A}=\mathbb{H}\,$, or the subring $ \,\sum_{i=0}^7 \mathbb{Z} u_i\,$ in the case $ \,{\cal A}=\mathbb{O}\,$.

For a large positive parameter $ X$, let $ \,A(X)\,$ denote the number of squares $ \alpha^2$ with $ \,\alpha \in \Gamma\,$ and all $ s$ components of $ \alpha^2$ lying in the interval $ [-X,X]$. Then, generalizing former results concerning the distribution of squares of Gaussian integers by H. Müller and W.G. Nowak, we show that

$\displaystyle A(X)\;=\;c\,X^{s/2}\,-\,d\,X^{(s-1)/2}\,+\,
\hbox{\rm O}\left(
X(\log X)^{-1/2}+X^{(s-2)/2}\delta(X)\right)\;\;(\,X\to\infty\,)\,,$

where $ c$ and $ d$ are certain positive constants depending on $ \Gamma$, and $ \delta(X)$ is any upper bound of the error term in the divisor problem, e.g. $ \,\delta(X)=X^{0.315}$.

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

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