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Mertens equimodular matrices of Redheffer type. (English) Zbl 1440.11166

Summary: Asymptotic behavior of the Mertens function \(M(n) = \sum_{k = 0}^n \mu(k)\) which is equal to the determinant of the \(n \times n\) Redheffer matrix, is known to be closely related to the Riemann hypothesis. An infinite matrix whose \(n\)th leading principal minor is equal to \(M(n)\) for all \(n \geq 1\) is called a Mertens equimodular matrix. We use Riordan matrices to find a large class of Mertens equimodular matrices, each element of which is called by us a Riordan-Redheffer matrix, briefly an R-R matrix. We also give the generating function for the characteristic polynomials of R-R matrices. As a result, we introduce several examples of R-R matrices that reveal interesting spectral properties. Further, we pose two conjectures on the eigenvalues of those R-R matrices. Finally, we find a sufficient condition for the Riemann hypothesis using the smallest singular value of a R-R matrix.

MSC:

11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
15A18 Eigenvalues, singular values, and eigenvectors

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