We define generalised zeta functions associated with 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.

我们定义与签名\（（g-1,1）\）的不确定二次形式相关联的广义zeta函数，更笼统地说，是虚部具有签名\（（g-1,1）\）的复杂对称矩阵，并且我们调查它们的特性。在Zwegers的意义上，这些不确定的zeta函数定义为无限theta函数的Mellin变换，然后将其广义化为Siegel模块化设置。我们证明了不确定的zeta函数的解析连续性和泛函方程。我们还表明，维度2中的不确定zeta函数专用于实数二次域的射线类zeta函数的差异，根据Stark猜想，其在\（s = 0 \）处的领先泰勒系数预计为代数单位的对数。