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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.08527 |
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Table of Contents:
- We determine the smoothed counts of $S_4$-quartic fields with bounded discriminant, satisfying any finite specified set of local conditions, as the sum of two main terms with a power saving error term. We also prove an analogous result for quartic rings (weighted by the number of cubic resolvents), deducing as a consequence that the Shintani zeta functions associated to the prehomogeneous vector space $\mathbb{C}^2\otimes\mathrm{Sym}^2(\mathbb{C}^3)$ have at most a simple pole at $s=5/6$.