A sharp almost sure upper bound for partial sums of random multiplicative functions

Technical Summary: A Sharp Almost Sure Upper Bound for Partial Sums of Random Multiplicative Functions

Problem Statement
The paper addresses the almost sure asymptotic behavior of the partial sums $M_f(x) = \sum_{n \le x} f(n)$, where $f$ is a random multiplicative function. The study focuses on two specific models:

1. Steinhaus Model: $f(p)$ are independent and uniformly distributed on the unit circle $\mathbb{T}$, extended completely multiplicatively.
2. Rademacher Model: $f(p)$ are independent symmetric signs ($\pm 1$), extended multiplicatively to squarefree integers (supported on $\mu^2(n)=1$).

While the second moment $E|M_f(x)|^2$ scales as $x$, the almost sure upper bound is expected to be larger due to correlations. Previous work established a lower bound of order $\sqrt{x}(\log \log x)^{1/4}$ (Harper [12]) and various upper bounds with larger logarithmic exponents, such as $\sqrt{x}(\log \log x)^{2+\varepsilon}$ (Basquin [3]; Lau–Tenenbaum–Wu [13]) and $\sqrt{x}(\log \log x)^{3/4+\varepsilon}$ (Caich [5]). Harper conjectured that the sharp almost sure upper bound should match the lower bound up to an arbitrary $\varepsilon$ factor in the exponent, specifically $\sqrt{x}(\log \log x)^{1/4+\varepsilon}$.

Methodology
The authors employ a multi-stage probabilistic and analytic strategy to prove the conjectured bound. The proof follows the general architecture of Caich [5] but introduces a critical refinement in the moment estimation.

1. Discretization and Interpolation:
   The continuous variable $x$ is replaced by a sparse sequence of "test points" $x_i = \lfloor \exp(i c_0) \rfloor$. Lemma 3.8 demonstrates that the fluctuation of $M_f(x)$ between consecutive test points is negligible (smaller than any power of $\log x$) almost surely, allowing the problem to be reduced to bounding $M_f(x_i)$.

2. Largest-Prime Decomposition:
   The sum $M_f(x_i)$ is decomposed based on the largest prime factor $P(n)$ of the integers $n \le x_i$. The primes are partitioned into blocks $(y_{j-1}, y_j]$.

   • Smooth Term: Integers with $P(n) \le y_0$.
   • Single-Occurrence Term ($M_f^{(1)}$): Integers where the largest prime appears exactly once. This forms a prime-indexed martingale.
   • Repeated-Occurrence Term ($M_f^{(2)}$): Integers where the largest prime appears at least twice. In the Rademacher model, this term vanishes.

3. Smoothing and Quadratic Variation:
   The control of the single-occurrence term relies on bounding its predictable quadratic variation $V_\ell(x_i; f)$. The authors utilize a smoothing inequality (Proposition 3.15) to relate $V_\ell$ to a principal logarithmic mean square term $M_\ell$ and several auxiliary error terms ($\lambda$-terms, boundary terms, and $W$-terms). The auxiliary terms are shown to be summable via standard probabilistic tools (Doob's inequalities, Azuma–Hoeffding).

4. Key Innovation: Conditional Higher Moments:
   The core difficulty lies in controlling the principal term $M_\ell$ uniformly across the $J \asymp \log \log x$ prime blocks.

   • Previous Approach (Caich): Relied on conditional first-moment estimates summed over blocks, resulting in a loss of a factor of $\sqrt{\log \log x}$.
   • Current Approach: The authors introduce a fixed conditional $r$-th moment estimate (Proposition 5.2 for Steinhaus, Proposition 7.8 for Rademacher). They define a tilted, normalized Euler product integral $I_j$ and prove it forms a supermartingale (Lemma 5.1, Lemma 7.7).
   • Crucially, they establish that $E[U_j^r \mid \mathcal{F}_{y_{j-1}}] \ll I_{j-1}^r$, where $U_j$ dominates the principal term. This higher-moment control, combined with a stopping argument (using the $2/3$-moment bound from Harper [9] for the Steinhaus case and a shifted estimate for Rademacher), yields a compensating factor of $L^{-1}$ (where $L \asymp \log \log x$). This precisely cancels the union bound over the $J$ blocks, eliminating the extra $\sqrt{\log \log x}$ loss.

5. Model-Specific Adjustments:

   • Steinhaus: Uses the standard hypercontractive inequality (Lemma 3.1) and Harper's low-moment estimate for the full-line mean square (Lemma 4.1).
   • Rademacher: Requires modifications because signs are not rotationally invariant. The authors use a shifted unit-interval estimate (Lemma 7.4) and adapt the supermartingale construction to the specific Euler product structure of the Rademacher model (Lemma 7.7).

Key Results
The main theorem establishes the sharp almost sure upper bound for both models:

Theorem 1: Let $f$ be either a Steinhaus or Rademacher random multiplicative function. For every $\varepsilon > 0$, almost surely:
$$|M_f(x)| \ll_{\varepsilon, f} \sqrt{x} (\log \log x)^{1/4 + \varepsilon}$$

This is supported by Proposition 1.1, which provides a more precise fixed-order statement: for any integer $r \ge 2$ and $\eta > 0$, almost surely:
$$|M_f(x)| \ll_{r, \eta, f} \sqrt{x} (\log \log x)^{1/4 + 1/(2r) + \eta}$$

Combining this with Harper's lower bound (1.6), the authors derive Corollary 1.2, determining the exact logarithmic exponent:
$$\limsup_{x \to \infty} \frac{\log(1 + |M_f(x)|/\sqrt{x})}{\log \log \log x} = \frac{1}{4}$$

Significance and Claims
The paper claims to settle Harper's conjecture regarding the large fluctuations of random multiplicative functions. By determining the sharp logarithmic exponent of $1/4$, the work resolves the discrepancy between previous upper bounds (which had exponents like $2$ or $3/4$) and the known lower bound.

The authors emphasize that the proof does not rely on independence arguments (which fail due to multiplicative correlations) but rather on a refined understanding of how these correlations persist as $x$ varies. The primary technical contribution is the replacement of the conditional first-moment step in prior work with a fixed conditional $r$-th moment estimate on a single prime block. This innovation allows for the precise absorption of the union bound over the logarithmic number of prime blocks, achieving the conjectured exponent. The paper asserts that this result, together with the lower bound, fully characterizes the almost sure size of partial sums for these random multiplicative functions.