### 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 language | English |
---|---|

Pages (from-to) | 637-662 |

Number of pages | 26 |

Journal | Publicationes Mathematicae |

Volume | 79 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 2011 |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Publicationes Mathematicae*,

*79*(3-4), 637-662. https://doi.org/10.5486/PMD.2011.5068

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

Research output: Contribution to journal › Article

*Publicationes Mathematicae*, vol. 79, no. 3-4, pp. 637-662. https://doi.org/10.5486/PMD.2011.5068

}

TY - JOUR

T1 - A Turán-Kubilius type inequality on shifted products

AU - Rivat, Joäel

AU - Sárközy, A.

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

KW - Additive arithmetic function

KW - Large sieve

KW - Multiplicative characters

KW - Probabilistic number theory

KW - Shifted product

UR - http://www.scopus.com/inward/record.url?scp=84867523113&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867523113&partnerID=8YFLogxK

U2 - 10.5486/PMD.2011.5068

DO - 10.5486/PMD.2011.5068

M3 - Article

AN - SCOPUS:84867523113

VL - 79

SP - 637

EP - 662

JO - Publicationes Mathematicae

JF - Publicationes Mathematicae

SN - 0033-3883

IS - 3-4

ER -