The distribution of sequences in residue classes

Christian Elsholtz*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review


    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→ ∞.

    Original languageEnglish
    Pages (from-to)2247-2250
    Number of pages4
    JournalProceedings of the American Mathematical Society
    Issue number8
    Publication statusPublished - Aug 2002


    • 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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