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Register a new userA new formula for Chebotarev densities
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Authors
Madeline Locus Dawsey
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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 $ngeq2$. 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 $$-lim_{Xrightarrowinfty}sum_{substack{2leq nleq X[1pt]left[frac{K/mathbb{Q}}{p_{mathrm{min}}(n)}right]=C}}frac{mu(n)}{n}=frac{#C}{#G}.$$ 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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