Non-Abelian Generalizations of the Erdős-Kac Theorem

Murty, M. Ram; Saidak, Filip

doi:10.4153/cjm-2004-017-7

Cited by 15 publications

(16 citation statements)

References 17 publications

(21 reference statements)

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“…Under GRH the following result was proved by Saidak in his PhD thesis and Ram Murty and Saidak [378]. Let a ≥ 2 be an integer.…”

Section: ) Erdős-kac Type Theorems For the Multiplicative Ordermentioning

confidence: 88%

Artin's Primitive Root Conjecture – A Survey

Moree

2012

Integers

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Abstract. One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the literature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contributions in the survey on 'elliptic Artin' are due to Alina Cojocaru. Wojciec Gajda wrote a section on 'Artin for K-theory of number fields', and Hester Graves (together with me) on 'Artin's conjecture and Euclidean domains'.

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“…Under GRH the following result was proved by Saidak in his PhD thesis and Ram Murty and Saidak [378]. Let a ≥ 2 be an integer.…”

Section: ) Erdős-kac Type Theorems For the Multiplicative Ordermentioning

confidence: 88%

Artin's Primitive Root Conjecture – A Survey

Moree

2012

Integers

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“…We prove these results of Ω(F a (m)) in Section 5 to conclude the paper. Our approach in Section 4 is different from the ones in [3] and [15]. In previous works, the equivalences between Theorems 4 and 6, and their analogues for Ω(F a (m)), are proved independently from one another.…”

Section: Theorem 6 For a Monic Polynomial M ∈ A Let A And F A (M) Bmentioning

confidence: 97%

“…The first breakthrough of the problem was recently achieved by Murty and Saidak [15]. Under the GRH, they proved that the conjecture is true.…”

Section: Introductionmentioning

confidence: 97%

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A Carlitz module analogue of a conjecture of Erdos and Pomerance

Kuo

Liu

2009

Trans. Amer. Math. Soc.

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Abstract. Let A = F q [T ] be the ring of polynomials over the finite field F q and 0 = a ∈ A. Let C be the A-Carlitz module. For a monic polynomial m ∈ A, let C(A/mA) andā be the reductions of C and a modulo mA respectively. Let f a (m) be the monic generator of the ideal {f ∈ A, C f (ā) =0} on C(A/mA). We denote by ω(f a (m)) the number of distinct monic irreducible factors of f a (m). If q = 2 or q = 2 and a = 1, T , or (1 + T ), we prove that there exists a normal distribution for the quantityThis result is analogous to an open conjecture of Erdős and Pomerance concerning the distribution of the number of distinct prime divisors of the multiplicative order of b modulo n, where b is an integer with |b| > 1, and n a positive integer.

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“…Erdős and Kac proved this theorem by a probabilistic idea, building upon the work of Hardy and Ramanujan ( [10]) and Turán ([21]) on the normal order of ω(n). Since then there has been a very rich literature on various aspects of the Erdős-Kac theorem (see, for example, [1,9,11,13,14,15,16,17,19,20]). Interested readers can refer to Granville and Soundararajan's paper [8] for the most recent account and Elliot's monograph [6] for a comprehensive treatment of the subject.…”

Section: Introductionmentioning

confidence: 99%