Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 25020, 6 pages
doi:10.1155/JIA/2006/25020
An upper bound for the ℒP norm of a GCD-related matrix
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.
Abstract
We find an upper bound for the ℓp 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<r−1/p, then ‖((i,j)s/[i,j]r)n×n‖p<ζ(rp)2/pζ(rp−sp)1/p/ζ(2rp)1/p for all positive integers n, where ζ is the Riemann zeta function.