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

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

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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