The distribution of sequences in residue classes

Christian Elsholtz

doi:10.1090/S0002-9939-02-06395-5

The distribution of sequences in residue classes

Christian Elsholtz^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that any set of integers A ⊂ [1, x] with |A| ≫ (logx)^r lies in at least va(p) ≫ p^r/r+1 many residue classes modulo most primes p ≪ (log x)r+1. (Here r is a positive constant.) This generalizes a result of Erdos and Ram Murty, who proved in connection with Artin's conjecture on primitive roots that the integers below x which are multiplicatively generated by the coprime integers a₁,..., a_r (i.e. whose counting function is also c(log x)^r) lie in at least p^r/r+1+ε(p) residue classes, modulo most small primes p, where ε(p) → O, as p→ ∞.

Original language	English
Pages (from-to)	2247-2250
Number of pages	4
Journal	Proceedings of the American Mathematical Society
Volume	130
Issue number	8
DOIs	https://doi.org/10.1090/S0002-9939-02-06395-5
Publication status	Published - Aug 2002

Keywords

Artin's conjecture
Distribution of sequences in residue classes
Gallagher's larger sieve
Primitive roots

ASJC Scopus subject areas

Mathematics(all)
Applied Mathematics

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10.1090/S0002-9939-02-06395-5

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abstract = "We prove that any set of integers A ⊂ [1, x] with |A| ≫ (logx)r lies in at least va(p) ≫ pr/r+1 many residue classes modulo most primes p ≪ (log x)r+1. (Here r is a positive constant.) This generalizes a result of Erdos and Ram Murty, who proved in connection with Artin's conjecture on primitive roots that the integers below x which are multiplicatively generated by the coprime integers a1,..., ar (i.e. whose counting function is also c(log x)r) lie in at least pr/r+1+ε(p) residue classes, modulo most small primes p, where ε(p) → O, as p→ ∞.",

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AB - We prove that any set of integers A ⊂ [1, x] with |A| ≫ (logx)r lies in at least va(p) ≫ pr/r+1 many residue classes modulo most primes p ≪ (log x)r+1. (Here r is a positive constant.) This generalizes a result of Erdos and Ram Murty, who proved in connection with Artin's conjecture on primitive roots that the integers below x which are multiplicatively generated by the coprime integers a1,..., ar (i.e. whose counting function is also c(log x)r) lie in at least pr/r+1+ε(p) residue classes, modulo most small primes p, where ε(p) → O, as p→ ∞.

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KW - Gallagher's larger sieve

KW - Primitive roots

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The distribution of sequences in residue classes

Abstract

Keywords

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