Indefinite zeta functions

TL;DR

This paper introduces a new class of indefinite zeta functions linked to indefinite quadratic forms and complex symmetric matrices, establishing their analytic properties and connections to number theory.

Contribution

It defines indefinite zeta functions via Mellin transforms of indefinite theta functions and proves their analytic continuation and functional equations, extending the theory to the Siegel modular setting.

Findings

Proved analytic continuation and functional equations for indefinite zeta functions.

Connected indefinite zeta functions to ray class zeta functions of real quadratic fields.

Predicted relations of Taylor coefficients at s=0 to algebraic units via Stark conjectures.

Abstract

We define generalised zeta functions associated to indefinite quadratic forms of signature (g-1,1) -- and more generally, to complex symmetric matrices whose imaginary part has signature (g-1,1) -- and we investigate their properties. These indefinite zeta functions are defined as Mellin transforms of indefinite theta functions in the sense of Zwegers, which are in turn generalised to the Siegel modular setting. We prove an analytic continuation and functional equation for indefinite zeta functions. We also show that indefinite zeta functions in dimension 2 specialise to differences of ray class zeta functions of real quadratic fields, whose leading Taylor coefficients at s=0 are predicted to be logarithms of algebraic units by the Stark conjectures.