Dirichlet series associated to representation numbers of ideals in real quadratic number fields

TL;DR

This paper explicitly computes representation numbers of ideals in real quadratic fields to express associated Dirichlet series via L-functions and divisor sums, revealing their meromorphic continuation and pole structure.

Contribution

It provides explicit formulas for representation numbers and Dirichlet series in real quadratic fields, connecting them to L-functions and divisor sums for the first time.

Findings

Dirichlet series has meromorphic continuation to the entire complex plane.

The series has a simple pole at s=2 with an explicit residue.

Representation numbers are explicitly computed for real quadratic ideals.

Abstract

In this rather computational paper, we determine certain representation numbers of ideals in real quadratic number fields explicitly in order to obtain a representation of the associated Dirichlet series in terms of Dirichlet L-functions and a generalized divisor sum. A direct and important consequence is that the Dirichlet series has a meromorphic continuation to the whole complex plane and a simple pole at $s=2$ whose residue can be made explicit in terms of the Dirichlet L-functions and the generalized divisor sum.