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 δ(S) within the primes, thenwhere µ(a) is the generalized Möbius function and D(K, S) is the set of integral ideals a ⊆ OK 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 ⌊πn⌋.

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