On the eigenstructure of sparse matrices related to the prime number theorem

The sparse (0,1) matrix that was introduced by Redheffer has the property that its nth leading principal minor equals ∑i≤nμ(i), where μ is the Möbius function. Recently, a substantially sparser (0,1) matrix that possesses the same property has been described. Let Rn denote the nth leading principal submatrix of this sparser matrix. Our main result establishes asymptotically tight estimates for the two dominant eigenvalues of Rn. We also state explicit formulae for the eigenvectors of Rn that are associated with the eigenvalues that are different from 1. Finally, we state several conjectures about the eigenstructure of a related class of matrix.