A Kronecker limit formula for indefinite zeta functions

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...

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Bibliographic Details
Published in:Research in the mathematical sciences 2023-06, Vol.10 (2), Article 24
Main Author: Kopp, Gene S.
Format: Article
Language:English
Subjects:
Applications of Mathematics
Computational Mathematics and Numerical Analysis
Fields (mathematics)
Mathematical analysis
Mathematics
Mathematics and Statistics
Mellin transforms
Number theory
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Summary: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.
ISSN:2522-0144
2197-9847
DOI:10.1007/s40687-023-00384-0