Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 25020, 6 pages

An upper bound for the P norm of a GCD-related matrix

Pentti Haukkanen

Department of Mathematics, Statistics and Philosophy, University of Tampere, Tampere 33014, Finland

Received 10 November 2004; Revised 12 January 2005; Accepted 9 February 2005

Copyright © 2006 Pentti Haukkanen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We find an upper bound for the p norm of the n×n matrix whose ij entry is (i,j)s/[i,j]r, where (i,j) and [i,j] are the greatest common divisor and the least common multiple of i and j and where r and s are real numbers. In fact, we show that if r>1/p and s<r1/p, then ((i,j)s/[i,j]r)n×np<ζ(rp)2/pζ(rpsp)1/p/ζ(2rp)1/p for all positive integers n, where ζ is the Riemann zeta function.