A dominant negative eigenvalue of a matrix of Redheffer (Q752138)

The paper is concerned with a matrix \(A_ n\) introduced by \textit{R. M. Redheffer} [Eine explizit lösbare Optimierungsaufgabe. ISNM 36, 213-216 (1977)], which is connected with the Riemann hypothesis via Merten's function. Particularly, the existence of a real eigenvalue of \(A_ n\) is proven which is asymptotically equal to -\(\sqrt{n}\). It is also shown that the eigenvalues, except \(\rho (A_ n)\), the spectral radius, and the eigenvalue of order -\(\sqrt{n}\), are contained in the circle \(\{\) \(z\in C:| z-1| <\rho (A_ n)/\log n\}\) if n is sufficiently large.