Möbius formulas for densities of sets of prime ideals

Kural, Michael; McDonald, Vaughan; Sah, Ashwin

doi:10.48550/arxiv.1907.02914

Cited by 2 publications

(7 citation statements)

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“…Hence, by (44) we finally obtain (37) and the conclusion (7). This completes the proof of Corollary 1.2.…”

Section: Proof Of Corollary 12supporting

confidence: 74%

“…Then Sweeting and Woo [10] generalized Dawsey's result to number fields. Recently, Kural et al [7] generalized all these results to natural densities of sets of primes, see section 2. Over Q, the result of Kural et al is stated as follows.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 97%

“…For the proof of Theorem 1.1, we follow the approach of proving Theorem 1 in [14]. Then we apply Theorem 1.1 to show (7). For ( 8)-( 10), we will explain how to derive them from (7) in section 2.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…In this section, we give two examples of natural densities and explain how they are used to derive the analogues of Alladi's and Dawsey's results from (6) or (7). We refer interested readers to [7] for more interesting examples. To apply Theorem 1.1 and Corollary 1.2, notice that…”

Section: Examples Of Natural Densitiesmentioning

confidence: 99%

“…At the end of this note, we give some examples that are analogous to (7) or (13) for the ϕ(n) by replaced by functions "near n". We may call them "mock n-functions".…”

Section: Examplesmentioning

confidence: 99%

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Analogues of Alladi's formula

Wang

2020

Preprint

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In this note, we mainly show the analogue of one of Alladi's formulas over Q with respect to the Dirichlet convolutions involving the Möbius function µ(n), which is related to the natural densities of sets of primes by recent work of Dawsey, Sweeting and Woo, and Kural et al. This would give us several new analogues. In particular, we get that if (k, ℓ) = 1, thenwhere p(n) is the smallest prime divisor of n, and ϕ(n) is Euler's totient function. This refines one of Hardy's formulas in 1921. At the end, we give some examples for the ϕ(n) replaced by functions "near n", which include the sum-of-divisors function.

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“…Hence, by (44) we finally obtain (37) and the conclusion (7). This completes the proof of Corollary 1.2.…”

Section: Proof Of Corollary 12supporting

confidence: 74%

Section: Introduction and Statement Of Resultsmentioning

confidence: 97%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Examples Of Natural Densitiesmentioning

confidence: 99%

“…At the end of this note, we give some examples that are analogous to (7) or (13) for the ϕ(n) by replaced by functions "near n". We may call them "mock n-functions".…”

Section: Examplesmentioning

confidence: 99%

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