Almost sure upper bound for random multiplicative functions

Rachid Caich

Almost sure upper bound for random multiplicative functions

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arxiv 2304.00943 v2 pith:UKK36SKG submitted 2023-04-03 math.NT math.PR

Rachid Caich This is my paper · ORCID

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keywords almostmultiplicativerandomvarepsilonboundfracfunctionfunctions

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abstract

Let $\varepsilon >0$. Let $f$ be a Steinhaus or Rademacher random multiplicative function. We prove that we have almost surely, as $x \to +\infty$, $$ \sum_{n \leqslant x} f(n) \ll \sqrt{x} (\log_2 x)^{\frac{3}{4}+ \varepsilon}. $$

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A sharp almost sure upper bound for partial sums of random multiplicative functions
math.NT 2026-07 conditional novelty 8.0 of 10

For both Steinhaus and Rademacher random multiplicative functions, almost surely |sum_{n≤x} f(n)| ≪ sqrt(x)(log log x)^{1/4+ε}, matching Harper's lower bound.

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