A Turán-Kubilius type inequality on shifted products

Joäel Rivat, A. Sárközy

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

In 1934 Turán proved that if f(n) is an additive arithmetic function satisfying certain conditions, then for almost all m ≤ n the value of f(m) is "near" the expectation ∑ p≤n f(p)/p. Later Kubilius sharpened this result by proving that the conditions in Turán's theorem can be relaxed, and still the same conclusion holds. In an earlier paper we studied whether this result has a sum set analogue, i.e., if f(n) is an additive arithmetic function and A, B are "large" subsets of {1,2,...,n}, then for almost all a ∈ A, b ∈ B, the value of f(a + b) is "near" the expectation? We proved such a result under an assumption which is slightly milder than Turán's condition, but is not needed in Kubilius estimate. In this paper we prove the multiplicative analogue of this theorem by proving a similar result with ab + 1 in place of a + b.

Original languageEnglish
Pages (from-to)637-662
Number of pages26
JournalPublicationes Mathematicae
Volume79
Issue number3-4
DOIs
Publication statusPublished - 2011

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Arithmetic Functions
Additive Function
Analogue
Sumsets
Theorem
Multiplicative
Subset
Estimate

Keywords

  • Additive arithmetic function
  • Large sieve
  • Multiplicative characters
  • Probabilistic number theory
  • Shifted product

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

A Turán-Kubilius type inequality on shifted products. / Rivat, Joäel; Sárközy, A.

In: Publicationes Mathematicae, Vol. 79, No. 3-4, 2011, p. 637-662.

Research output: Contribution to journalArticle

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