Redheffer's matrix

The Redheffer matrix $A(n)$ is an $n\times n$ matrix of 0's and 1's defined by $A(i,j) = 1$ if $j = 1$ or if $i$ divides $j$, and $A(i,j) = 0$ otherwise. Redheffer proved that $A(n)$ has $ n-[n \log 2]-1$ eigenvalues equal to 1. Also, A has a real eigenvalue (the spectral radius) which is approximately $\sqrt(n)$, a negative eigenvalue which is approximately $-\sqrt{n}$ and the remaining eigenvalues are small. The Riemann hypothesis is true if and only if $\det(A) = O(n^{1/2+\epsilon})$ for every $\epsilon > 0$.

Barrett Forcade, Rodney, and Pollington [ MR 89j:15029] give an easy proof of Redheffer's theorem. They also prove that the spectral radius of $A(n)$ is $=n^{1/2}+\tfrac 12\log n+O(1).$ See also the paper of Roesleren [ MR 87i:11111].

Vaughan [ MR 94b:11086] and [ MR 96m:11073] determines the dominant eigenvalues with an error term $O(n^{-2/3})$ and shows that the nontrivial eigenvalues are $\ll(\log n)^{2/5}$ (unconditionally), and $\ll\log\log(2+n)$ on Riemann's hypothesis.

It is possible that the nontrivial eivenvalues lie in the unit disc.

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