Two-Variable Zeta Functions and Regularized Products

In this paper we prove a regularized product expansion for the two-variable zeta functions of number fields introduced by van der Geer and Schoof. The proof is based on a general criterion for zeta-regularizability due to Illies. For number fields of non-zero unit rank our method involves a result of independent interest about the asymptotic behaviour of certain oscillatory integrals in the geometry of numbers. We also explain the cohomological motivation for the paper.

2000 Mathematics Subject Classification: 11M38; 11M41; 11H41; 14G40

Keywords and Phrases: zeta function, zeta regularization, oscillatory integral, metrized lattice

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