Low moments of sums of extended Rademacher random multiplicative functions

@misc{atherfold2026moments,
  title = {{Low moments of sums of extended Rademacher random multiplicative functions}},
  author = {Christopher Atherfold and Maxim Gerspach and Jad Hamdan},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05492},
  eprint = {2611.05492},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05492},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Low moments of sums of extended Rademacher random multiplicative functions
AU  - Christopher Atherfold
AU  - Maxim Gerspach
AU  - Jad Hamdan
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.05492
UR  - https://arxiv.org/abs/2611.05492
AB  - Let X^e : ℕ → {± 1} be the random, completely multiplicative function whose values at the primes (X^e(p))_p prime are independent and uniformly distributed on { ± 1 }. Let X=X^eμ^2. We determine the order of magnitude of 𝔼 [ | 1/√x∑_n ≤ x X^e(n) |^2 q ] and 𝔼 [ |∑_n ≤ x X(n)/√n |^2 q ] for q∈ [0,1], showing that a transition from better-than-square-root cancellation to growth like a power of log x occurs when q crosses 1/2. In particular, the first absolute moments are of order (loglog x)^3/4. The proof proceeds by comparing these sums to integrals of random Euler products, which are then studied by analogy with Gaussian multiplicative chaos integrals. The main difficulty lies in proving the lower bound at q=1/2, for which we also adapt arguments from the study of random walks.
ER  -