A new formula for Chebotarev densities

Abstract

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 $p_{\min }(n)$ of integers $n\ge 2$. 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 $$-\underset{X\to \infty }{\lim }\sum _{\begin{array}{c}2\le n\le X\\ \left[\frac{K/\mathbb{Q}}{p_{\min }(n)}\right]=C\end{array}}\frac{\mu (n)}{n}=\frac{\#C}{\#G}.$$ This theorem is a generalization of a result of Alladi from 1977 that asserts that largest prime divisors $p_{\max }(n)$ are equidistributed in arithmetic progressions modulo an integer $k$, which occurs when $K$ is a cyclotomic field $\mathbb{Q}({\zeta }_k)$.

Cited by in corpus (1)

• Eulerian series, zeta functions and the arithmetic of partitions