On the largest singular vector of the Redheffer matrix

Abstract

The Redheffer matrix $A_n\in R^{n\times n}$ 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 $\det \left(A_n\right)=O\left(n^{1/2+\epsilon }\right)$ is equivalent to the Riemann hypothesis. The singular vector $v\in R^n$ corresponding to the largest singular value carries a lot of information about the prime factorization of the integers: $v_k$ 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.