Möbius formulas for densities of sets of prime ideals
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Author(s)
Kural, Michael
McDonald, Vaughan
Sah, Ashwin
Date Issued
April 29, 2020
Publisher
Springer International Publishing
Version
Author's final manuscript
Abstract
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 $$\delta (S)$$δ(S) within the primes, then $$\begin{aligned} -\lim _{X \rightarrow \infty }\sum _{\begin{array}{c} 2 \le {\text {N}}(\mathfrak {a})\le X\\ \mathfrak {a} \in D(K,S) \end{array}}\frac{\mu (\mathfrak {a})}{{\text {N}}(\mathfrak {a})} = \delta (S), \end{aligned}$$-limX→∞∑2≤N(a)≤Xa∈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⊆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 a Beatty sequence such as $$\lfloor \pi n\rfloor $$⌊πn⌋.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00013-020-01458-z
