On the largest singular vector of the Redheffer matrix
François Clément, Stefan Steinerberger
TL;DR
The paper investigates the top singular vector of the Redheffer matrix $A_n$ by proposing the divisor-sum vector $v_n=(\sum_{d|k}1/d)_{k\le n}$ as a close proxy. It develops a gcd-based surrogate $B_n=(\sigma_0(\gcd(i,j)))_{i,j\le n}$ and derives sharp asymptotics for $\|v_n\|$, $B_n v_n$, and $\langle v_n, B_n v_n\rangle$, enabling a precise comparison with $A_n^T A_n$. The main result establishes the existence of a limit $\alpha$ for the normalized overlap $\left\langle \tfrac{v_n}{\|v_n\|}, \tfrac{A_n^T A_n v_n}{\|A_n^T A_n v_n\|} \right\rangle$, given by an explicit closed form in terms of Dirichlet-convolution constants $c_d$ and $\zeta(3)$, with numerical value $\alpha\approx 0.997992$. This quantifies how closely $v_n$ approximates a top singular vector and reveals a deep connection between the spectral structure and number-theoretic constructs such as gcd-based matrices and Ramanujan identities. Overall, the work provides a rigorous, quantitative bridge between the largest singular vector of $A_n$ and a natural arithmetic vector, highlighting the influence of primes and highly composite indices on the spectrum.
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.