Partial sums of random multiplicative functions with supercritical divisor twists

1. Problem Statement and Context

The paper investigates the statistical behavior of partial sums of Steinhaus random multiplicative functions ($f$) twisted by the $\alpha$-divisor function ($d_\alpha$). Specifically, it analyzes the $2q$-th moments of the normalized sum:
$$S_x = \frac{1}{\sqrt{x}} \sum_{n \le x} d_\alpha(n) f(n)$$
where $f(n)$ is defined by independent random variables $Z_p$ on the complex unit circle for primes $p$, extended completely multiplicatively.

The Core Challenge:

• Critical Case ($\alpha = 1$): Harper (2016) resolved Helson's conjecture, showing that for $\alpha=1$, the moments exhibit "surprising cancellation" compared to independent random variables. The moments scale as $(\log \log x)^{-q/2}$ (roughly), a phenomenon explained by the theory of Critical Gaussian Multiplicative Chaos (GMC).
• Supercritical Case ($\alpha \in (1, 2)$): This paper addresses the regime where the twist $d_\alpha$ introduces a "supercritical" phase. In this regime, the standard critical GMC intuition fails, and the behavior is governed by the supercritical GMC theory. The goal is to establish sharp upper bounds for the moments of these twisted sums, matching predictions from supercritical GMC theory.

2. Methodology

The author develops a new, self-contained approach to compare the moments of the partial sums to the moments of integrals of random Euler products. The methodology avoids the complex "approximate Girsanov's theorem" used in previous works (like Harper's) and instead relies on the large deviation theory of random Euler products.

Key Technical Steps:

1. Reduction to Euler Products:
   Using sieve-theoretic arguments and Parseval's theorem, the author reduces the problem of bounding the moments of the discrete sum $\sum d_\alpha(n)f(n)$ to bounding the moments of the integral of the associated Euler product $F_y(s) = \prod_{p \le y} (1 - f(p)p^{-s})^{-1}$:
   $$\mathbb{E}\left[ \left| \frac{1}{\sqrt{x}} \sum_{n \le x} d_\alpha(n)f(n) \right|^{2q} \right] \approx \mathbb{E}\left[ \left( \frac{1}{\log y} \int_0^1 |F_y(\sigma + ih)|^{2\alpha} dh \right)^q \right]$$
   where $\sigma$ is close to $1/2$.

2. Analysis of Log-Correlated Fields:
   The logarithm of the Euler product, $\log |F_y(\sigma + ih)|$, behaves like a log-correlated Gaussian field (a random walk with logarithmic correlations). The author studies the measure of level sets of this field.

   • Critical Case ($\gamma=2$): The integral is dominated by points where the field is near its typical maximum ($\log \log x$).
   • Supercritical Case ($\gamma > 2$): The integral is dominated by the local maxima of the field. The analysis requires precise control over the maximum of the field and the measure of points near these maxima.

3. Level Set Measure and Maxima:
   The paper introduces a "good set" $G_A$ of points $h$ where the random walk $S_j(h)$ stays within specific upper and lower barriers.

   • Upper Barriers: The author constructs precise barriers $U_A(j)$ that account for the "logarithmic bump" typical of log-correlated fields (specifically the $-\frac{3}{4} \log \log x$ correction).
   • Discretization and Chaining: Using chaining arguments and discretization of the interval $[0, 1]$, the author bounds the probability that the field exceeds these barriers.
   • Tilted Measures: To handle the supercritical case ($\gamma > 2$), the author employs a tilted measure argument (change of measure) to analyze the behavior of the field near its maximum, effectively shifting the drift to study the "typical" contribution to the integral.

4. Interpolation Argument:
   The proof utilizes an interpolation technique (inspired by Soundararajan and Zaman) to combine estimates for different barrier heights, allowing the derivation of the final moment bounds for all $q$ in the relevant range.

3. Key Contributions and Results

Theorem 1.1 (Main Result: Supercritical Moments)

For $\alpha \in (1, 2)$ and $q \in [0, 1/\alpha]$, the paper establishes the following uniform bound for large $x$:
$$\mathbb{E}\left[ \left| \frac{1}{\sqrt{x}} \sum_{n \le x} d_\alpha(n)f(n) \right|^{2q} \right] \ll \frac{(\log x)^{2q(\alpha-1)}}{(\log \log x)^{3\alpha q/2 (1-\alpha q)} + 1}$$

• Significance: This matches the predictions from supercritical Gaussian Multiplicative Chaos. It confirms that the "double-logarithmic" correction $(\log \log x)$ is present but with a different exponent structure than the critical case.

Short Proof of Helson's Conjecture ($\alpha=1$)

The author provides a new, shorter proof for the critical case ($\alpha=1$) by applying the same level-set measure techniques. This recovers Harper's result:
$$\mathbb{E}\left[ \left| \frac{1}{\sqrt{x}} \sum_{n \le x} f(n) \right|^{2q} \right] \ll (\sqrt{\log \log x}(1-q) + 1)^{-q}$$
This proof bypasses the need for the approximate Girsanov theorem, simplifying the existing literature.

Theorem 1.3 (Moments of Euler Products)

A general bound for the moments of the integral of the Euler product raised to a power $\gamma \in (2, 4)$:
$$\mathbb{E}\left[ \left( \frac{1}{\log y} \int_0^1 |F_y(\sigma_k + ih)|^\gamma dh \right)^q \right] \ll \frac{(\log y)^{(\gamma-2)q}}{(\log \log y)^{3\gamma q/4 (2-\gamma q)} + 1}$$
This result is crucial as it isolates the behavior of the random Euler product itself, independent of the divisor function twist.

Corollaries on Extrema (1.4 & 1.5)

• Maximum Bound: The paper derives a sharp bound for the maximum of the Euler product on the critical line, matching the best-known bounds for the Riemann zeta function (Fyodorov-Hiary-Keating conjecture context).
• Typical Measure of Level Sets: It quantifies the measure of points where the Euler product is near its maximum, showing that these points are rare but dominate the integral in the supercritical regime.

Theorem 1.6 (Pseudomoments of the Riemann Zeta Function)

The results are applied to the pseudomoments of the Riemann zeta function, $\Psi_{2q, \alpha}(x)$. The paper proves an upper bound for the regime $1 \le \alpha < 2$ and small $q$:
$$\Psi_{2q, \alpha}(x) \ll \frac{(\log x)^{2q(\alpha-1)}}{(\log \log x)^{q(3\alpha/2)}}$$
This answers a question posed by Gerspach, confirming the conjectured exponent for the $(\log \log x)$ factor in this parameter range.

4. Significance

1. Unification of Critical and Supercritical Regimes: The paper successfully bridges the gap between the critical ($\alpha=1$) and supercritical ($\alpha > 1$) phases of random multiplicative functions, showing that the same underlying mechanism (log-correlated fields and level set measures) governs both, albeit with different scaling behaviors.
2. Methodological Advancement: By replacing the complex probabilistic machinery (Girsanov's theorem) with a direct analysis of level sets and large deviations, the author provides a more transparent and potentially more flexible toolkit for studying random multiplicative functions.
3. Resolution of Open Problems: It settles the asymptotic behavior of pseudomoments for the Riemann zeta function in a specific parameter range, a problem that had remained open despite significant prior work by Gerspach, Arguin, and others.
4. Connection to GMC: The results provide strong evidence for the universality of Gaussian Multiplicative Chaos in number theory, demonstrating that the "supercritical" phase of GMC accurately models the behavior of divisor-twisted random multiplicative functions.

In summary, Hamdan's work offers a rigorous, self-contained, and technically refined framework for understanding the extreme values and moments of random multiplicative functions, extending seminal results from the critical case to the supercritical regime and resolving specific conjectures regarding the Riemann zeta function.