Bordered Hermitian matrices and sums of the Möbius function.

Matrices in R n × n with determinant equal to ∑ i ≤ n μ (i) , where μ represents the Möbius function, have been studied for decades. The motivation to study such matrices is the close connection that they have to the prime number theorem, Dirichlet convolution, and related concepts. We introduce two parameterized families of bordered Hermitian matrices that possess similar properties. Each family is comprised of matrices M s ∈ C (n − 1) × (n − 1) that satisfy det ⁡ M 0 = ∑ i ≤ n | μ (i) | , det ⁡ M 1 = (∑ i ≤ n μ (i)) 2 , and we show det ⁡ M s is a quadratic polynomial in s. We apply 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 / π 2 + O (n − 1 / 2). [ABSTRACT FROM AUTHOR]