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Indefinite zeta functions. (English) Zbl 1470.11058

Summary: 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.

MSC:

11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)
11F27 Theta series; Weil representation; theta correspondences
11M41 Other Dirichlet series and zeta functions
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