The Redheffer matrix is an matrix of 0's and 1's defined by if or if divides , and otherwise. Redheffer proved that has eigenvalues equal to 1. Also, A has a real eigenvalue (the spectral radius) which is approximately , a negative eigenvalue which is approximately and the remaining eigenvalues are small. The Riemann hypothesis is true if and only if for every .
Barrett Forcade, Rodney, and Pollington [ MR 89j:15029] give an easy proof of Redheffer's theorem. They also prove that the spectral radius of is 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 and shows that the nontrivial eigenvalues are (unconditionally), and on Riemann's hypothesis.
It is possible that the nontrivial eivenvalues lie in the unit disc.
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