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 $$\delta (S)$$δ(S) within the primes, then $$\begin{aligned} -\lim _{X \r...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Springer International Publishing
2021
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| Online Access: | https://hdl.handle.net/1721.1/131373 |
| _version_ | 1811082549010628608 |
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| author | Kural, Michael McDonald, Vaughan Sah, Ashwin |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Kural, Michael McDonald, Vaughan Sah, Ashwin |
| author_sort | Kural, Michael |
| collection | MIT |
| description | Abstract
We generalize results of Alladi, Dawsey, and Sweeting and Woo for Chebotarev densities to general densities of sets of primes. We show that if K is a number field and S is any set of prime ideals with natural density $$\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 $$⌊πn⌋. |
| first_indexed | 2024-09-23T12:05:10Z |
| format | Article |
| id | mit-1721.1/131373 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T12:05:10Z |
| publishDate | 2021 |
| publisher | Springer International Publishing |
| record_format | dspace |
| spelling | mit-1721.1/1313732023-02-22T17:22:47Z Möbius formulas for densities of sets of prime ideals Kural, Michael McDonald, Vaughan Sah, Ashwin Massachusetts Institute of Technology. Department of Mathematics Abstract We generalize results of Alladi, Dawsey, and Sweeting and Woo for Chebotarev densities to general densities of sets of primes. We show that if K is a number field and S is any set of prime ideals with natural density $$\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⌋. 2021-09-20T17:16:47Z 2021-09-20T17:16:47Z 2020-04-29 2020-09-24T21:09:55Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/131373 en https://doi.org/10.1007/s00013-020-01458-z Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer Nature Switzerland AG application/pdf Springer International Publishing Springer International Publishing |
| spellingShingle | Kural, Michael McDonald, Vaughan Sah, Ashwin Möbius formulas for densities of sets of prime ideals |
| title | Möbius formulas for densities of sets of prime ideals |
| title_full | Möbius formulas for densities of sets of prime ideals |
| title_fullStr | Möbius formulas for densities of sets of prime ideals |
| title_full_unstemmed | Möbius formulas for densities of sets of prime ideals |
| title_short | Möbius formulas for densities of sets of prime ideals |
| title_sort | mobius formulas for densities of sets of prime ideals |
| url | https://hdl.handle.net/1721.1/131373 |
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