Analogues of Alladi's formula

Wang, Biao

doi:10.1016/j.jnt.2020.06.001

Cited by 9 publications

(18 citation statements)

References 9 publications

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“…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. More precisely, Wang [27] proved that if an arithmetic function a : N → C satisfies a(1) = 1 and…”

Section: Introduction and Main Resultsmentioning

confidence: 97%

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Generalizations of Alladi's formula for arithmetical semigroups

Duan,

Ma,

2021

Preprint

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Section: Introduction and Main Resultsmentioning

confidence: 97%

“…

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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confidence: 99%

Generalizations of Alladi's formula for arithmetical semigroups

Duan,

Ma,

2021

Preprint

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“…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].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

A "supernormal" partition statistic

Dawsey¹,

Just²,

Schneider³

2021

Preprint

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We study a natural bijection from integer partitions to the prime factorizations of integers that we call the "supernorm" of a partition, in which the multiplicities of the parts of partitions are mapped to the multiplicities of prime factors of natural numbers. In addition to being inherently interesting, the supernorm statistic suggests the potential to apply techniques from partition theory to identify and prove multiplicative properties of integers. We prove an analogue of a formula of Kural-McDonald-Sah: we give arithmetic densities of subsets of N instead of natural densities in P like previous formulas of this type, as an application of the supernorm, building on works of Alladi, Ono, Wagner and the first and third authors. We then make an analytic study of related partition statistics; and suggest speculative applications based on the additive-multiplicative analogies we prove, to conjecture Abeliantype formulas. In an appendix, using these same analogies, we give a conjectural partition-theoretic model of prime gaps.

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“…Over the last few years, (1.2) has been generalized in various ways in works by Dawsey [Da17], Sweeting and Woo [SwWo19], Kural, McDonald and Sah [KMS], and Wang [Wa1,Wa2]. In a recent preprint [Wa3], Wang obtained a beautiful generalization of this phenomenon which makes use of Dirichlet convolutions. If a : N → C is an arithmetic function satisfying a(1) = 1 and In earlier work the authors obtained a partition-theoretic analogue of formulas such as (1.2) using natural analogies between the multiplicative structure of the integers and the additive structure of partitions.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Now we turn our attention toward proving the partition convolution identities analogous to formulas of Wang [Wa3]. It is an instance of (2.12) for arbitrary partition-theoretic functions a(λ), b(λ)…”

Section: Proof Of Theorem 13 and Its Corollaries 4a Partition Convolution Identitiesmentioning

confidence: 98%

Partition-theoretic formulas for arithmetic densities, II

Ono

Schneider²,

Wagner

2021

Hardy-Ramanujan Journal

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International audience In earlier work generalizing a 1977 theorem of Alladi, the authors proved a partition-theoretic formula to compute arithmetic densities of certain subsets of the positive integers N as limiting values of q-series as q → ζ a root of unity (instead of using the usual Dirichlet series to compute densities), replacing multiplicative structures of N by analogous structures in the integer partitions P. In recent work, Wang obtains a wide generalization of Alladi's original theorem, in which arithmetic densities of subsets of prime numbers are computed as values of Dirichlet series arising from Dirichlet convolutions. Here the authors prove that Wang's extension has a partition-theoretic analogue as well, yielding new q-series density formulas for any subset of N. To do so, we outline a theory of q-series density calculations from first principles, based on a statistic we call the "q-density" of a given subset. This theory in turn yields infinite families of further formulas for arithmetic densities.

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Analogues of Alladi's formula

Cited by 9 publications

References 9 publications

Generalizations of Alladi's formula for arithmetical semigroups

Generalizations of Alladi's formula for arithmetical semigroups

A "supernormal" partition statistic

Partition-theoretic formulas for arithmetic densities, II

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