A sparser matrix representation of the Mertens function (Q2273884)

It is well known that the growth of the Mertens function is closely related to the localization of the zeros of the Riemann zeta function. In this context, the tasks of interest are: generating a sparse \((0, 1)\) matrix whose determinant provide an unorthodox representation of the Mertens function [\textit{R. Redheffer}, Tag. Oberwolfach 1976, ISNM 36, 213--216 (1977; Zbl 0363.65062)] as well as the study of the spectral properties of a matrix of Redheffer-type [\textit{W. W. Barrett} and \textit{T. J. Jarvis}, Linear Algebra Appl. 162--164, 673--683 (1992; Zbl 0746.15009)]. The author studies a \((0,1)\) matrix whose determinant also provides a nontrivial representation of the Mertens function, but it is sparser than the matrix of Redheffer. More precisely, the matrix has fewer nonzero entries (\(O(n)\) compared to \(O(n \log n)\)), and it has substantially fewer eigenvalues different from 1. A sparser family of matrices are discussed. Some empirical results about the matrices are presented in Section 3.