Lau 2013 mean values random multiplicative functions

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theorem_1_1: Lau, Tenenbaum and Wu's almost-sure bound for the partial sums M_f(x) of a random multiplicative function supported on squarefree integers: for every epsilon > 0, almost surely M_f(x) << x^{1/2} (log_2 x)^{3/2+epsilon} as x tends to infinity.

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Lau, Yuk-Kam and Tenenbaum, Gérald and Wu, Jie, On mean values of random multiplicative functions. Proc. Amer. Math. Soc. 141 (2013), no. 2, 409-420, DOI 10.1090/S0002-9939-2012-11332-2. The copy read for this card is the authors' manuscript deposited as HAL record hal-01278413; its HAL cover sheet (p. 1) prints "HAL Authorization", and HAL's own record names the HAL authorization v1 as the deposit's license, the depositor's authorization for HAL to distribute it, with no public reuse grant and no Creative Commons line (HAL API record, read 2026-10-02; the record page could not be read on 2026-10-02), every other right reserved.

Let f be the random multiplicative function supported on squarefree integers built from independent Bernoulli signs f(p) = +-1, and M_f(x) the sum of f(n) for n <= x. Theorem 1.1 proves that for every epsilon > 0 one has almost surely M_f(x) << x^{1/2} (log_2 x)^{3/2+epsilon}, in a slightly more general model where f(p) vanishes with probability 1 - kappa_p subject to a prime-sum condition (1.8). This qualitatively matches the law of the iterated logarithm for genuinely independent signs and improves Halasz's bound x^{1/2} exp(c_4 sqrt(log_2 x log_3 x)), which itself had improved Wintner's x^{1/2+epsilon} and Erdos's logarithmic refinement. The method follows Halasz's approach with new refinements (compare their Lemma 3.1 with Lemma 3(ii) of Halasz's paper) that remove the log_3 x factor and more. Harper's lower bound (1.6), M_f(x) >> x^{1/2}/(log_2 x)^{5/2+epsilon} almost surely for infinitely many x, shows how narrow the remaining gap is. For problem 520 the theorem is an almost-sure upper bound on the same sums, though its exponent 3/2 + epsilon on log_2 x is above the exponent 1/2 of the problem's normalization (x log_2 x)^{1/2}, so it does not decide the question.

The manuscript is dated 30 June 2011 and paginated 1--10; labels and pages on this card and its result pages are the manuscript's.

Read status: claims checked for the result linked below, its statement read clause by clause on the printed pages; no proof is checked step by step.

Source: https://hal.science/hal-01278413v1.

Bears on.

• #520: the problem's Rademacher function is the case kappa_p = 1 of the paper's model, and Theorem 1.1 gives almost surely sum_{m <= N} f(m) << N^{1/2} (log_2 N)^{3/2+epsilon}; the exponent 3/2 + epsilon exceeds the 1/2 of the question's normalization (N log_2 N)^{1/2}, so the theorem does not answer the question.

Results.

• Theorem 1.1 (p. 3): For every epsilon > 0, almost surely M_f(x) << x^{1/2}(log_2 x)^{3/2+epsilon} as x -> infinity, in the model (1.7) with P(f(p)=0) = 1 - kappa_p, where kappa_p in [0,1] satisfies (1.8). The context the paper recalls on p. 2, Halasz's bound (1.4) and Harper's lower bound (1.6), is stated on that result page.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.