PreprintCC BY 4.0arXiv
Large fluctuations of extended Rademacher random multiplicative functions
Abstract
Let f be an extended Rademacher random multiplicative function (RMF). We show that, for every fixed deterministic function V(x) tending to infinity, almost surely there are arbitrarily large x for which ∑n≤ xf(n) ≥ (√x(loglog x)1/4)/V(x). The corresponding negative fluctuation holds as well. In particular, this gives an affirmative answer to Erdős Problem~\#1144. As a byproduct, our result has a direct corollary giving new almost sure lower bounds loglog x on the number of sign changes of partial sums up to x for all sufficiently large x.
Related
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BibTeX
@misc{gou2026large,
title = {{Large fluctuations of extended Rademacher random multiplicative functions}},
author = {Haozhe Gou and Max Wenqiang Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12940},
eprint = {2609.12940},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12940},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Large fluctuations of extended Rademacher random multiplicative functions
AU - Haozhe Gou
AU - Max Wenqiang Xu
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.12940
UR - https://arxiv.org/abs/2609.12940
AB - Let f be an extended Rademacher random multiplicative function (RMF). We show that, for every fixed deterministic function V(x) tending to infinity, almost surely there are arbitrarily large x for which ∑_n≤ xf(n) ≥ (√x(loglog x)^1/4)/V(x). The corresponding negative fluctuation holds as well. In particular, this gives an affirmative answer to Erdős Problem~\#1144. As a byproduct, our result has a direct corollary giving new almost sure lower bounds loglog x on the number of sign changes of partial sums up to x for all sufficiently large x.
ER -