On the largest singular vector of the Redheffer matrix

Abstract

The Redheffer matrix $A_n\in {\mathbb{R}}^{n\times n}$ is defined by setting $A_{ij}=1$ if $j=1$ or $i$ divides $j$ and 0 otherwise. One of its many interesting properties is that $\det (A_n)=O(n^{1/2+\epsilon })$ is equivalent to the Riemann hypothesis. The singular vector $v\in {\mathbb{R}}^n$ corresponding to the largest singular value carries a lot of information: $v_k$ is small if $k$ is prime and large if $k$ has many divisors. We prove that the vector $w$ whose $k-$th entry is the sum of the inverse divisors of $k$, $w_k={\sum }_{d|k}1/d$, is close to a singular vector in a precise quantitative sense.