A sharp almost sure upper bound for partial sums of random multiplicative functions

@misc{durkan2026sharp,
  title = {{A sharp almost sure upper bound for partial sums of random multiplicative functions}},
  author = {Benjamin Durkan and Andrew Pearce-Crump},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.29429},
  eprint = {2607.29429},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.29429},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - A sharp almost sure upper bound for partial sums of random multiplicative functions
AU  - Benjamin Durkan
AU  - Andrew Pearce-Crump
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2607.29429
UR  - https://arxiv.org/abs/2607.29429
AB  - We prove that, for either a Steinhaus random multiplicative function or a Rademacher random multiplicative function f, and every ε>0, almost surely |∑_n≤ xf(n)|≪_ε,f√x(loglog x)^1/4+ε. Together with Harper's almost sure lower bound, this determines the sharp logarithmic exponent in both models. This settles Harper's conjecture on the large fluctuations of random multiplicative functions.
ER  -