The Powers of Certain Number-Theoretic Matrices

Abstract

For each integer $n\ge 3$ we define a specific symmetric integer matrix of size $\phi (n)/2$. We then explicitly describe all positive and negative integer powers, as well as fractional powers, of each of these matrices; their eigenvalues also appear and play an important role. All this is based on an orthogonality relation for odd Dirichlet characters which, in turn, leads to a diagonalization. In the course of this article, various other number-theoretic objects make their appearance, including the Euler and Jordan totient functions, Möbius inversion, and the divisor function.

MSC::

• Primary 11C20
• Secondary 11A25
• 15B36

ACKNOWLEDGMENT

We wish to thank the anonymous referee for a careful reading of this article and for helpful suggestions.

Additional information

Funding

Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada;

Notes on contributors

Karl Dilcher

KARL DILCHER received his undergraduate education at the Technische Universität Clausthal in Germany. He then did his graduate studies at Queen’s University in Kingston, Ontario, and finished his Ph.D. there in 1983. He is currently a professor at Dalhousie University in Halifax, Nova Scotia, Canada, where he first arrived in 1984 as a postdoctoral fellow. His research interests include special functions and number theory.

Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, B3H 4R2, Canada

[email protected]

Kurt Girstmair

KURT GIRSTMAIR received his Ph.D. in 1978 from Innsbruck University. After two years as a Humboldt research fellow in Germany he became an Assistant Professor and then an Associate Professor in Innsbruck. He is retired now. His main research interests are algebra and number theory.

Institut für Mathematik, Universität Innsbruck, Technikerstr. 13/7, A-6020 Innsbruck, Austria

[email protected]