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Formulas for Chebotarev densities of Galois extensions of number fields

  • Naomi Sweeting
  • Katharine WooEmail author
Research
  • 37 Downloads

Abstract

We generalize the Chebotarev density formulas of Dawsey (Res Number Theory 3:27, 2017) and Alladi (J Number Theory 9:436–451, 1977) to the setting of arbitrary finite Galois extensions of number fields L / K. In particular, if \(C \subset G = {{\mathrm{Gal}}}(L/K)\) is a conjugacy class, then we establish that the Chebotarev density is the following limit of partial sums of ideals of K:
$$\begin{aligned} -\lim _{X\rightarrow \infty } \sum _{\begin{array}{c} 2\le N(I)\le X \\ I \in S(L/K; C) \end{array}} \frac{\mu _K(I)}{N(I)} = \frac{|C|}{|G|}, \end{aligned}$$
where \(\mu _K(I)\) denotes the generalized Möbius function and S(L / KC) is the set of ideals \(I\subset \mathcal {O}_K\) such that I has a unique prime divisor \(\mathfrak {p}\) of minimal norm and the Artin symbol \(\left[ \frac{L/K}{\mathfrak {p}}\right] \) is C. To obtain this formula, we generalize several results from classical analytic number theory, as well as Alladi’s concept of duality for minimal and maximal prime divisors, to the setting of ideals in number fields.

Notes

Author's contributions

Acknowlegements

The authors would like to thank Professor Ken Ono and Professor Larry Rolen for their guidance and suggestions. They also thank Emory University, the Asa Griggs Candler Fund, and NSF grant DMS-1557960.

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Copyright information

© SpringerNature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Department of MathematicsStanford UniversityStanfordUSA

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