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Lithuanian Mathematical Journal

, Volume 59, Issue 1, pp 24–38 | Cite as

Distribution of Arithmetic Functions on Particular Subsets of Integers

  • Jean-Marie De KoninckEmail author
  • Imre Kátai
Article
  • 25 Downloads

Abstract

Let Q1, . . . , Qt[x] be polynomials with no constant term for which each linear combination m1Q1(x) + · · ·+mtQt (x) with m1, . . . , mtand not all 0 always has an irrational coefficient. Let I1, . . . , It be sets included in the interval [0, 1), each being a union of finitely many subintervals of [0, 1). Furthermore, let \( \mathcal{T} \) be the set of positive integers n for which {Q1(n)} ∈ I1, . . . , {Qt(n)} ∈ It simultaneously, where {y} stands for the fractional part of y. Let t1, t2, . . . be a uniformly summable sequence of complex numbers and set T(x) =  ∑ n ≤ xtn and\( T\left(x|\mathcal{T}\right)=\sum {}_{n\le x,n\in \mathcal{T}}{t}_n \). We prove that, as x→∞, \( T(x)/x\sim T\left(x|\ \mathcal{T}\right)/\left(\uplambda \left({I}_1\right)\cdots \uplambda \left({I}_t\right)x\right) \), where λ(I) is the Lebesgue measure of a set I.

Keywords

polynomials primes arithmetic function 

MSC

11K16 11N37 11A41 

Notes

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Département de mathématiques et de statistiqueUniversité Laval, QuébecQuébecCanada
  2. 2.Computer Algebra DepartmentEötvös Loránd UniversityBudapestHungary

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