On the largest singular vector of the Redheffer matrix

TL;DR

This paper investigates the properties of the largest singular vector of the Redheffer matrix, revealing its relation to divisor sums and prime numbers, and connects these findings to the Riemann hypothesis.

Contribution

It establishes a quantitative approximation of the largest singular vector by the divisor sum vector, linking matrix spectral properties to number theory.

Findings

The vector of inverse divisor sums closely approximates the largest singular vector.

The entries of the singular vector reflect the divisor structure of integers.

Connections are drawn between spectral properties and prime number distribution.

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 $\det(A_n) = O(n^{1/2 + \varepsilon})$ 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.