Caich 2023 almost sure upper bound random multiplicative

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Rachid Caich, Almost sure upper bound for random multiplicative functions. arXiv:2304.00943 (2023). The arXiv record (https://arxiv.org/abs/2304.00943, read 2026-10-02) names the Creative Commons Attribution 4.0 license. The folder's PDF is arXiv:2304.00943v2 [math.NT] (19 August 2024; 26 pages), whose pagination is used here.

For f a Steinhaus or Rademacher random multiplicative function, Theorem 1.1 proves that almost surely M_f(x) = sum_{n<=x} f(n) << sqrt(x) (log_2 x)^{3/4 + eps} for every fixed eps > 0 (p. 3). This improves the almost sure bound sqrt(x)(log_2 x)^{2+eps} that Lau, Tenenbaum and Wu proved in the Rademacher case, which itself refined Halász's Rademacher bound sqrt(x) exp(A sqrt(log_2 x log_3 x)) and earlier work of Wintner and Erdős (p. 2). The method sums over sparse test points, applies Borel--Cantelli, splits M_f according to the largest prime factor P(n) into a smooth part (P(n) <= y_0), a part in which P(n) divides n exactly once (a sum of martingale differences over the ranges y_{j-1} < p <= y_j, with J of order log_2 x ranges) and a part in which P(n) divides n at least twice (zero in the Rademacher case) (p. 3), and controls them with martingale and Doob/Hoeffding inequalities plus Harper's low-moment machinery. The paper also recalls Harper's lower bound, that for any V(x) tending to infinity, almost surely |M_f(x)| >> sqrt(x)(log_2 x)^{1/4}/V(x) for arbitrarily large x, and Harper's conjecture that almost surely M_f(x) << sqrt(x)(log_2 x)^{1/4+eps} for every fixed eps > 0 (pp. 2--3). This is progress towards Erdős problem 520, which asks whether the limsup of M_f(N)/sqrt(N log log N) is almost surely a positive constant: the new upper bound is (log_2 x)^{3/4+eps} rather than the (log_2 x)^{1/2} that such a law of the iterated logarithm would require, so it narrows but does not close the gap.

Source: https://arxiv.org/abs/2304.00943.

Bears on. #520

Results to transcribe.

• Theorem 1.1: For Steinhaus or Rademacher f and any eps > 0, almost surely sum_{n<=x} f(n) << sqrt(x)(log_2 x)^{3/4+eps} as x -> infinity.