Extending Redheffer's Matrix to Arbitrary Arithmetic FunctionsGillespie, Bryan RaeRedheffer matrixMertens functionRedheffer-type matrixarithmetic functionsembeddinggeneral linear groupDirichlet convolutionconvolution inversionsumseigenvaluescharactersvolumeEulerian polynomialsMathematicsB SThe 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.Robert Charles Vaughan, Thesis SupervisorRobert Charles Vaughan, Thesis SupervisorSvetlana Katok, Thesis Honors Advisoropen_access2011-04-18T21:32:12Zhttps://honors.libraries.psu.edu/catalog/1740