# M\"obius formulas for densities of sets of prime ideals

## TL;DR

This paper extends formulas for prime ideal densities in number fields using Möbius functions, providing new tools to analyze the distribution of primes in various sets, including those related to elliptic curves and Beatty sequences.

## Contribution

It generalizes Chebotarev density results to arbitrary prime sets in number fields using Möbius formulas, enabling new density calculations.

## Key findings

- Derived a Möbius formula for densities of prime sets in number fields.
- Applied the formula to sets of primes in elliptic curve Sato-Tate intervals.
- Extended the approach to primes in Beatty sequences.

## Abstract

We generalize results of Alladi, Dawsey, and Sweeting and Woo for Chebotarev densities to general densities of sets of primes. We show that if $K$ is a number field and $S$ is any set of prime ideals with natural density $\delta(S)$ within the primes, then \[ -\lim_{X \to \infty}\sum_{\substack{2 \le \operatorname{N}(\mathfrak{a})\le X\\ \mathfrak{a} \in D(K,S)}}\frac{\mu(\mathfrak{a})}{\operatorname{N}(\mathfrak{a})} = \delta(S), \] where $\mu(\mathfrak{a})$ is the generalized M\"obius function and $D(K,S)$ is the set of integral ideals $ \mathfrak{a} \subseteq \mathcal{O}_K$ with unique prime divisor of minimal norm lying in $S$. Our result can be applied to give formulas for densities of various sets of prime numbers, including those lying in a Sato-Tate interval of a fixed elliptic curve, and those in Beatty sequences such as $\lfloor\pi n\rfloor$.