Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 15 min ago
PreprintCC BY 4.0arXiv

A law of the iterated logarithm for random multiplicative functions on integers with a large prime factor

  • Dariusz Buraczewski
  • Alexander Iksanov
Open at sourcePDFDOICite

Abstract

Let f be a Rademacher random multiplicative function supported on the squarefree integers. We investigate the almost sure fluctuations of the restricted sum M>(x)=∑n≤ x; P(n)>x1/2f(n), where P(n) is the largest prime factor of n. With Q(x) being the conditional variance of M>(x) given the values of f at primes not exceeding x1/2, we prove that, almost surely, the upper and lower limits of M>(x)(2Q(x)loglog Q(x))−1/2 as x→∞ are $1$ and −1, respectively. Furthermore, it is shown that the almost sure cluster set of the normalized sum is [−1,1].

Related

  1. A self-normalized law of the iterated logarithm for Rademacher random multiplicative functions
    Dariusz Buraczewski, Alexander Iksanov
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryProbabilityarXiv
  2. Laws of the iterated logarithm for the range of once-reinforced random walk on ℤ
    Ze-Chun Hu, Renming Song, Li Wang
    arXivPreprintCC BY 4.0Full textProbabilityarXiv
  3. Common random number simulation as the optimal coordinatewise coupling for iterated random functions
    Sabrina Sixta, Jeffrey S. Rosenthal
    arXivPreprintarXiv licenceFull textStatisticsarXiv
  4. Almost sure upper bound for random multiplicative functions
    Rachid Caich
    Acta ArithmeticaPeer-reviewedCC BYFull text15 citedAlgebra & number theoryOpenAlex
  5. A sharp almost sure upper bound for partial sums of random multiplicative functions
    Benjamin Durkan, Andrew Pearce-Crump
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryProbabilityarXiv

Cite

BibTeXRIS
BibTeX
@misc{buraczewski2026iterated,
  title = {{A law of the iterated logarithm for random multiplicative functions on integers with a large prime factor}},
  author = {Dariusz Buraczewski and Alexander Iksanov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40707},
  eprint = {2610.40707},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40707},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A law of the iterated logarithm for random multiplicative functions on integers with a large prime factor
AU  - Dariusz Buraczewski
AU  - Alexander Iksanov
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.40707
UR  - https://arxiv.org/abs/2610.40707
AB  - Let f be a Rademacher random multiplicative function supported on the squarefree integers. We investigate the almost sure fluctuations of the restricted sum M^>(x)=∑_n≤ x; P(n)>x^1/2f(n), where P(n) is the largest prime factor of n. With Q(x) being the conditional variance of M^>(x) given the values of f at primes not exceeding x^1/2, we prove that, almost surely, the upper and lower limits of M^>(x)(2Q(x)loglog Q(x))^−1/2 as x→∞ are $1$ and −1, respectively. Furthermore, it is shown that the almost sure cluster set of the normalized sum is [−1,1].
ER  -