A new formula for Chebotarev densities

TL;DR

This paper introduces a novel formula for Chebotarev densities using smallest prime factors of integers, generalizing previous results and providing a new perspective on Frobenius element distribution in Galois groups.

Contribution

It presents a new formula for Chebotarev densities expressed via smallest prime factors, extending Alladi's 1977 results to general Galois extensions.

Findings

New formula for Chebotarev densities involving smallest prime factors

Generalization of Alladi's 1977 result on prime divisor distribution

Connection between prime factorization and Galois group conjugacy classes

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 $p_{\mathrm{min}}(n)$ 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 $\mathbb{Q}(\zeta_k)$.