The Divisor Matrix, Dirichlet Series and SL(2,Z)

TL;DR

This paper constructs a novel representation of SL(2,Z) acting on Dirichlet series, linking the Riemann zeta function to algebraic equations through group actions and matrix representations.

Contribution

It introduces a new matrix-based representation of SL(2,Z) acting on Dirichlet series, connecting group actions with the Riemann zeta function.

Findings

Representation of SL(2,Z) on Dirichlet series constructed

Unipotent element acts as multiplication by zeta function

Dirichlet series in the orbit satisfy algebraic equations

Abstract

A representation of SL(2,Z) by integer matrices acting on the space of analytic ordinary Dirichlet series is constructed, in which the standard unipotent element acts as multiplication by the Riemann zeta function. It is then shown that the Dirichlet series in the orbit of the zeta function are related to it by algebraic equations.