Arithmetic Functions and Integer Products

Elliott, P. D. T. A.

doi:10.1007/978-1-4613-8548-6

Cited by 62 publications

(62 citation statements)

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“…Furthermore, y 3 = x 3 , for otherwise we obtain the equality (2ak + 1)(α + y 4 ) = α + x 4 , which is impossible as α / ∈ Q. Similarly, 4 is not a permutation of y 1 , y 2 , y 3 , y 4 , which is the desired conclusion.…”

Section: Proofsmentioning

confidence: 81%

Multiplicative dependence of shifted algebraic numbers

Drungilas

Dubickas

2003

Colloq. Math.

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Abstract. We show that the set obtained by adding all sufficiently large integers to a fixed quadratic algebraic number is multiplicatively dependent. So also is the set obtained by adding rational numbers to a fixed cubic algebraic number. Similar questions for algebraic numbers of higher degrees are also raised. These are related to the ProuhetTarry-Escott type problems and can be applied to the zero-distribution and universality of some zeta-functions.

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Section: Proofsmentioning

confidence: 81%

Multiplicative dependence of shifted algebraic numbers

Drungilas

Dubickas

2003

Colloq. Math.

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“…There is a well-developed theory with many general results about the existence of means of arithmetic functions, see Elliott (1985); Indlekofer (1980Indlekofer ( , 1981Postnikov (1988). However, those general results do not imply the specific statements of this work.…”

mentioning

confidence: 83%

Average order in cyclic groups

Gathen¹,

Knopfmacher²,

Luca³

et al. 2004

J. Théor. Nombres Bordeaux

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Résumé. Pour chaque entier naturel n, nous déterminons l'ordre moyen α(n) deséléments du groupe cyclique d'ordre n. Nous montrons que plus de la moitié de la contributionà α(n) provient des ϕ(n)éléments primitifs d'ordre n. Il est par conséquent intéressant d'étudierégalement la fonction β(n) = α(n)/ϕ(n). Nous détermi-nons le comportement moyen de α, β, 1/β et considérons aussi ces fonctions dans le cas du groupe multiplicatif d'un corps fini.Abstract. For each natural number n we determine the average order α(n) of the elements in a cyclic group of order n. We show that more than half of the contribution to α(n) comes from the ϕ(n) primitive elements of order n. It is therefore of interest to study also the function β(n) = α(n)/ϕ(n). We determine the mean behavior of α, β, 1/β, and also consider these functions in the multiplicative groups of finite fields.

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“…More recently, Bombieri, Friedlander and Iwaniec [2] (see also [1]), using an idea of Motohashi [9], gave a very general Bombieri-Vinogradov type theorem for functions / that can be represented as convolutions of "well-behaved" functions. Elliott [3,Chapter 7] [4] and the author [8] showed that all additive functions satisfy a Bombieri-Vinogradov type theorem.…”

Section: ■* (Aq)=\mentioning

confidence: 99%