Möbius formulas for densities of sets of prime ideals (1907.02914v2)

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 $\lfloor\pi n\rfloor$.