Möbius formulas for densities of sets of prime ideals

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 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⌋.

Generalizations of Alladi's formula for arithmetical semigroups

“…The result (1) was generalized by Dawsey [5] to the setting of Chebotarev densities for finite Galois extensions of Q, and further by Sweeting and Woo [22] to number fields. Later, Kural et al [17] generalized all these results to natural densities of sets of prime ideals of number field K. Let P be the set of prime ideals p ⊆ O K and we say that a subset S ⊆ P with natural density δ(S) if the following limit exists: A recent work [8] of Wang with the first and third authors of this article showed the analogue of Kural et al's result over global function fields. In the other direction, Wang [25,26,27] showed the analogues of these results over Q for some arithmetic functions other than µ.…”

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In this article, we prove that a general version of Alladi's formula with Dirichlet convolution holds for arithmetical semigroups satisfying Axiom A or Axiom A # . As applications, we apply our main results to certain semigroups coming from algebraic number theory, arithmetical geometry and graph theory, particularly generalizing the results of [27,17,8].

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A "supernormal" partition statistic

“…Other authors [14,27,28] have since extended Alladi's techniques from [1], giving a variety of what we call "Alladi-like" formulas of the shape n≥2 µ * (n)f (n)/n < ∞, with µ * (n) := −µ(n) and f (n) an arithmetic function, to compute natural densities and other constants, such as this special 1 For more in-depth treatments of the partition norm, see [12,22,25]. Partitions of fixed norm n are called multiplicative partitions of n (or factorizations of n) in the literature; these were apparently first studied by MacMahon [15].…”

“…Following Oppenheim [19], the study of these objects is sometimes referred to as "factorisatio numerorum". case of a result of Kural-McDonald-Sah [14] stated as [27], equation ( 4):…”

“…Moreover, it preserves stochastic heuristics on prime distribution in that partitions are also treated stochastically. 14 From the standpoint of scientific modeling, it is a pretty good "first-draft" model of the prime numbers. The reasonable predictive success of the model provides support for this appendix's central heuristic: the problem of prime gaps can be viewed as a problem about the enumeration of partitions.…”