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

TL;DR

This paper explores the connection between a specific elliptic curve derived from Dirichlet series representations of SL(2,Z), modular functions, and the Riemann zeta function, revealing new algebraic and modular relationships.

Contribution

It provides explicit descriptions of modular functions associated with the elliptic curve and details the permutation action of SL(2,Z) on these functions, linking them to the Riemann zeta function.

Findings

The elliptic curve is a modular curve for (15).

A certain orbit of modular functions is associated with the Riemann zeta function.

Explicit descriptions of these functions and their permutation actions are provided.

Abstract

We examine an elliptic curve constructed in an earlier paper from a certain representation of $\SL(2,\Z)$ on the space of convergent Dirichlet series. The curve is observed to be a modular curve for $\Gamma^1(15)$ and a certain orbit of modular functions is thereby associated with the Riemann zeta function. Explicit descriptions are given of these functions and of the permutation action of $\SL(2,\Z)$ on them.