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 $-\lim_{X\rightarrow\infty}\sum_{\substack{2\leq n\leq X\\[1pt]\left[\frac{K/\mathbb{Q}}{p_{\mathrm{min}}(n)}\right]=C}}\frac{μ(n)}{n}=\frac{\#C}{\#G}.$ 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 $\mathbb{Q}(ζ_k)$.

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