PreprintCC BY 4.0arXiv
Almost sure upper bound for sums of random multiplicative functions and critical chaos
Abstract
Let f be a Steinhaus or Rademacher random multiplicative function. We use methods from the theory of critical chaos to improve on the best known upper bound for partial sums of random multiplicative functions. In particular, our results imply that for any ε>0, almost surely |∑n≤ xf(n)| ≪ε√x(log2x)1/4(log3x)1+ε. This proves in a strong form a conjecture of Harper on large fluctuations of partial sums of random multiplicative functions, and determines the exact corresponding logarithmic exponent.
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BibTeX
@misc{verreault2026almost,
title = {{Almost sure upper bound for sums of random multiplicative functions and critical chaos}},
author = {William Verreault},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.21354},
eprint = {2608.21354},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.21354},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Almost sure upper bound for sums of random multiplicative functions and critical chaos
AU - William Verreault
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2608.21354
UR - https://arxiv.org/abs/2608.21354
AB - Let f be a Steinhaus or Rademacher random multiplicative function. We use methods from the theory of critical chaos to improve on the best known upper bound for partial sums of random multiplicative functions. In particular, our results imply that for any ε>0, almost surely |∑_n≤ xf(n)| ≪_ε√x(log_2x)^1/4(log_3x)^1+ε. This proves in a strong form a conjecture of Harper on large fluctuations of partial sums of random multiplicative functions, and determines the exact corresponding logarithmic exponent.
ER -