Algebraic Frameworks for Spectral Analysis
The study of metacyclic codes in group algebras provides a sophisticated entry point into the spectral theory of the Riemann zeta function. By leveraging the structure of non-abelian groups, specifically the semidirect product of cyclic groups, researchers can model the distribution of primes through the lens of multiplier automorphisms. This framework, explored in arXiv 2003.09347, utilizes the residue ring Rm = Fq[x]/(xm - 1) and its multiplier μr to establish a finite analogue of the Euler product.
Multiplier Dynamics and Spectral Operators
The multiplier operator μr, defined by the transformation of x to xr, acts as a permutation on the character group of a cyclic subgroup. In this model, the zeros of the zeta function are hypothesized to correspond to the eigenvalues of a self-adjoint operator constructed from the algebraic properties of the group structure.