Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A new formula for Chebotarev densities   We give a new formula for the Chebotarev densities of Frobenius elements in
Galois groups. This formula is given in terms of smallest prime factors
$p_{\mathrm{min}}(n)$ of integers $n\geq2$. More precisely, let $C$ be a
conjugacy class of the Galois group of some finite Galois extension $K$ of
$\mathbb{Q}$. Then we prove that
$$-\lim_{X\rightarrow\infty}\sum_{\substack{2\leq n\leq
X\\[1pt]\left[\frac{K/\mathbb{Q}}{p_{\mathrm{min}}(n)}\right]=C}}\frac{\mu(n)}{n}=\frac{\#C}{\#G}.$$
This theorem is a generalization of a result of Alladi from 1977 that asserts
that largest prime divisors $p_{\mathrm{max}}(n)$ are equidistributed in
arithmetic progressions modulo an integer $k$, which occurs when $K$ is a
cyclotomic field $\mathbb{Q}(\zeta_k)$.

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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: q-bio
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The categories of this paper are: cs
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The categories of this paper are: atom-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ex
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The categories of this paper are: solv-int
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The categories of this paper are: ao-sci
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The categories of this paper are: stat
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The categories of this paper are: dg-ga
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The categories of this paper are: physics
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ph
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The categories of this paper are: quant-ph