A Kronecker limit formula for indefinite zeta functions

TL;DR

This paper establishes a Kronecker-like limit formula for indefinite zeta functions, enabling insights into special values related to algebraic units and practical computation of Stark invariants.

Contribution

It introduces a new limit formula for indefinite zeta functions, extending classical results to higher dimensions and connecting to Stark's conjectures.

Findings

Derived a Kronecker limit formula for indefinite zeta functions in dimension 2.

Connected special values of these functions to algebraic units in real quadratic fields.

Enabled high-precision computation of Stark ray class invariants.

Abstract

We prove an analogue of Kronecker's second limit formula for a continuous family of "indefinite zeta functions". Indefinite zeta functions were introduced in the author's previous paper as Mellin transforms of indefinite theta functions, as defined by Zwegers. Our formula is valid in dimension g=2 at s=1 or s=0. For a choice of parameters obeying a certain symmetry, an indefinite zeta function is a differenced ray class zeta function of a real quadratic field, and its special value at $s=0$ was conjectured by Stark to be a logarithm of an algebraic unit. Our formula also permits practical high-precision computation of Stark ray class invariants.