Extending Redheffer's Matrix to Arbitrary Arithmetic Functions
Open Access
Author:
Gillespie, Bryan Rae
Area of Honors:
Mathematics
Degree:
Bachelor of Science
Document Type:
Thesis
Thesis Supervisors:
Robert Charles Vaughan, Thesis Supervisor Robert Charles Vaughan, Thesis Supervisor Svetlana Katok, Thesis Honors Advisor
Keywords:
Redheffer matrix Mertens function Redheffer-type matrix arithmetic functions embedding general linear group Dirichlet convolution convolution inversion sums eigenvalues characters volume Eulerian polynomials
Abstract:
The class of Redheffer matrices are distinctive for having determinants equal to the Mertens function. We describe an embedding of the arithmetic functions into the general linear group which allows a generalization of Redheffer's matrices. This generalization exhibits determinants equal to the sum of the Dirichlet convolution inverse of a given invertible arithmetic function, and allows a more general analysis of the mechanisms at work behind Redheffer's original matrices.
Following past work by Robert Vaughan, we conduct a basic but general analysis of
the Eigenvalues of these Redheffer-type matrices, and we conduct a more in-depth analysis on the matrices corresponding to non-principal Dirichlet characters. We additionally discuss an alternate geometric bound on an important class of coefficients encountered during the analysis, and present a natural generalization of the Eulerian polynomials which emerges when working
with certain convolution inverses.
