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.
Accessible Version in Progress
We're generating an accessible version of this file to meet ADA Title II requirements. This process may take up to one hour. Please return later to access the accessible copy once it's ready.
You can still download the current version by clicking "OK".
What's happening:
An accessible PDF is being generated using Adobe with AI used to generate alternative text (alt text) for images in the PDF.