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Shifted values of the largest prime factor function and its average value in short intervals

Volume 143 / 2016

Jean-Marie De Koninck, Imre Kátai Colloquium Mathematicum 143 (2016), 39-62 MSC: 11K06, 11N37. DOI: 10.4064/cm6474-12-2015 Published online: 3 December 2015

Abstract

We obtain estimates for the average value of the largest prime factor $P(n)$ in short intervals $[x,x+y]$ and of $h(P(n)+1)$, where $h$ is a complex-valued additive function or multiplicative function satisfying certain conditions. Letting $s_q(n)$ stand for the sum of the digits of $n$ in base $q\ge 2$, we show that if $\alpha $ is an irrational number, then the sequence $(\alpha s_q(P(n)))_{n\in \mathbb {N}}$ is uniformly distributed modulo 1.

Authors

  • Jean-Marie De KoninckDépartement de mathématiques et de statistique
    UniversitéLaval
    Québec G1V 0A6, Canada
    e-mail
  • Imre KátaiComputer Algebra Department
    Eötvös Loránd University
    Pázmány Péter Sétány I/C
    1117 Budapest, Hungary
    e-mail

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