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

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

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.