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 $$\delta (S)$$ δ ( S ) within the primes, then $$\begin{aligned} -\lim _{X \rightarrow \infty }\sum _{\begin{array}{c} 2 \le {\text {N}}(\mathfrak {a})\le X\\ \mathfrak {a} \in D(K,S) \end{array}}\frac{\mu (\mathfrak {a})}{{\text {N}}(\mathfrak {a})} = \delta (S), \end{aligned}$$ - lim X → ∞ ∑ 2 ≤ N ( a ) ≤ X a ∈ D ( K , S ) μ ( a ) N ( a ) = δ ( S ) , where $$\mu (\mathfrak {a})$$ μ ( a ) is the generalized Möbius function and D ( K , S ) is the set of integral ideals $$ \mathfrak {a} \subseteq \mathcal {O}_K$$ a ⊆ 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 a Beatty sequence such as $$\lfloor \pi n\rfloor $$ ⌊ π n ⌋ .

中文翻译：

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素理想集密度的莫比乌斯公式

我们将 Alladi、Dawsey、Sweeting 和 Woo 的 Chebotarev 密度结果推广到素数集的一般密度。我们证明，如果 K 是一个数域，S 是素数内具有自然密度 $$\delta (S)$$ δ ( S ) 的任何素理想集，那么 $$\begin{aligned} -\lim _{ X \rightarrow \infty }\sum _{\begin{array}{c} 2 \le {\text {N}}(\mathfrak {a})\le X\\ \mathfrak {a} \in D(K ,S) \end{array}}\frac{\mu (\mathfrak {a})}{{\text {N}}(\mathfrak {a})} = \delta (S), \end{aligned} $$ - lim X → ∞ ∑ 2 ≤ N ( a ) ≤ X a ∈ D ( K , S ) μ ( a ) N ( a ) = δ ( S ) ，其中 $$\mu (\mathfrak {a}) $$ μ ( a ) 是广义莫比乌斯函数，D ( K , S ) 是积分理想的集合 $$ \mathfrak {a} \subseteq \mathcal {O}_K$$ a ⊆ OK位于 S 中的范数。