Upper bounds for the condition numbers of the GCD and the reciprocal GCD matrices in spectral norm

Let S={x1,…,xn}S={x1,…,xn} be a set of nn distinct positive integers. The n×nn×n matrix having the greatest common divisor (xi,xj)(xi,xj) of xixi and xjxj as its i,ji,j-entry is called the greatest common divisor (GCD) matrix defined on SS, denoted by ((xi,xj))((xi,xj)), or abbreviated as (S)(S). The n×nn×n matrix (S−1)=(gij)(S−1)=(gij), where gij=1(xi,xj), is called the reciprocal greatest common divisor (GCD) matrix on SS. In this paper, we present upper bounds for the spectral condition numbers of the reciprocal GCD matrix (S−1)(S−1) and the GCD matrix (S)(S) defined on S={1,2,…,n}S={1,2,…,n}, with n≥2n≥2, as a function of Euler’s ϕϕ function and nn.