Bordered Hermitian matrices and sums of the Möbius function (Q2174446)

Real \(n\times n\) matrices with determinant equal to the Mertens function \(M(n)=\sum_{1\le i\le n}\mu(i)\) have been studied for decades, see, e.g., [\textit{R. Redheffer}, in: Numer. Meth. Optim.-Aufg., Band 3, Optim. graphentheor. ganzz. Probl., Tag. Oberwolfach 1976, ISNM 36, 213--216 (1977; Zbl 0363.65062)]. Here, \(\mu\) is the number-theoretic Möbius function. The motivation to study such matrices arises from their close connection to the prime number theorem and the Riemann hypothesis. In fact, the Riemann hypothesis is equivalent to \(M(n)=O(n^{1/2+\varepsilon})\). The present author introduces two parametrized families of bordered Hermitian matrices that possess similar properties. Each family is comprised of \((n-1)\times(n-1)\) matrices \(M_s\) such that \(\det M_s\) is a quadratic polynomial in \(s\). In addition, \(\det M_0 = \sum_{1\le i\le n}|\mu(i)|\) and \(\det M_1 = M^2(n)\). The author applies the Cauchy interlacing theorem to show that, for each matrix in one of the families, the product of all of the subdominant eigenvalues is bounded above by \(6/\pi^2 + O(n^{-1/2})\).