Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Formulas for Chebotarev densities of Galois extensions of number fields   We generalize the Chebotarev density formulas of Dawsey (2017) and Alladi
(1977) to the setting of arbitrary finite Galois extensions of number fields
$L/K$. In particular, if $C \subset G = \textrm{Gal}(L/K)$ is a conjugacy
class, then we establish that the Chebotarev density is the following limit of
partial sums of ideals of $K$: \[ -\lim_{X\rightarrow\infty}
\sum_{\substack{2\leq N(I)\leq X \\ I \in S(L/K; C)}} \frac{\mu_K(I)}{N(I)} =
\frac{|C|}{|G|}, \] where $\mu_K(I)$ denotes the generalized M\"obius function
and $S(L/K;C)$ 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.

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The categories of this paper are: chao-dyn
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The categories of this paper are: cs
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The categories of this paper are: ao-sci
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The categories of this paper are: funct-an
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The categories of this paper are: q-alg
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The categories of this paper are: comp-gas
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The categories of this paper are: atom-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: math-ph
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The categories of this paper are: q-bio
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The categories of this paper are: hep-lat
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The categories of this paper are: nlin
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The categories of this paper are: alg-geom
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The categories of this paper are: patt-sol
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The categories of this paper are: quant-ph