Low moments of random multiplicative functions twisted by Fourier coefficients of modular forms

This paper determines the order of magnitude of the $2q$-th moments of the sum of a random multiplicative function twisted by the Fourier coefficients of a fixed modular form for the range $0 \leq q \leq 1$.

The Gist

The Big Picture: Predicting the Chaos

Imagine you are standing in a massive, noisy stadium. Thousands of people are shouting different numbers at random. Some are shouting positive numbers, some negative, some complex numbers. You want to know: If you add up all these shouts, how loud will the total volume be?

In mathematics, this is a classic problem involving random multiplicative functions. These are sequences of numbers that behave like random noise but follow specific rules (multiplying them together works a certain way).

For a long time, mathematicians had a "rule of thumb" (a heuristic) for this problem. They thought: "If you have $x$ random numbers, the total sum should be roughly the square root of $x$." It's like saying if you flip a coin 100 times, the net difference between heads and tails will be around 10, not 100.

However, a mathematician named A. J. Harper recently discovered a twist. He found that for certain types of random sums, the total is actually smaller than the square root of $x$. It's as if the crowd is somehow coordinating to cancel each other out more efficiently than pure chance would suggest.

The New Discovery: Adding a "Flavor"

The authors of this paper, Peng Gao and Liangyi Zhao, asked a fascinating question: What happens if we add a specific "flavor" to the noise?

Instead of just random numbers, they took the random numbers and multiplied them by Fourier coefficients of Modular Forms.

• The Analogy: Imagine the random shouts are the background noise of the stadium. The "Fourier coefficients" are like a specific, complex melody played by a soloist.
• The Question: If the crowd shouts random numbers while the soloist plays this specific melody, does the total volume change? Does the crowd cancel out even more, or does the melody make the noise louder?

The Main Result: The "Sweet Spot"

The paper proves that even with this complex melody added to the random noise, the crowd still cancels out in the exact same surprising way Harper discovered.

The formula they derived looks like this:
$$\text{Total Volume} \approx \frac{x}{1 + (1-q)\sqrt{\log \log x}}$$

What does this mean in plain English?

1. The "Square Root" Myth: Usually, you'd expect the volume to be $\sqrt{x}$.
2. The Reality: The volume is actually slightly less than $\sqrt{x}$, but the amount it is less depends on a variable called $q$ (which represents how we measure the "loudness" or "moment" of the sum).
3. The "Log-Log" Factor: The term $\sqrt{\log \log x}$ is a very slow-growing number. It's like a whisper that gets slightly louder as the stadium gets bigger, but it takes a massive stadium to hear it clearly.

The authors show that whether the random noise is "Steinhaus" (shouting complex numbers on a circle) or "Rademacher" (shouting only +1 or -1), and whether the melody is from a standard modular form or a specific type, the cancellation effect remains the same.

Why is this a Big Deal?

Think of it like this:

• Old View: Randomness is chaotic. Sometimes things cancel out, sometimes they don't.
• Harper's View: Randomness has a hidden structure that makes it cancel out more than we thought.
• This Paper's View: Even if you mix that randomness with some of the most structured, beautiful, and complex objects in mathematics (Modular Forms), that hidden structure of cancellation persists.

It's like discovering that no matter what instrument you play in the background, the crowd in the stadium still manages to whisper in perfect unison, making the total noise quieter than anyone expected.

The "How" (The Magic Trick)

How did they prove this? They didn't just count numbers; they used a "probability microscope."

1. Breaking it Down: They broke the huge sum of numbers into smaller chunks based on the size of the prime numbers involved (like sorting the crowd by height).
2. Changing the Rules: They used a mathematical trick called "Girsanov's theorem" (think of it as changing the weather in the stadium). They temporarily changed the probability of the random numbers to make the math easier to calculate, calculated the result, and then changed the weather back.
3. The Gaussian Connection: They showed that the behavior of these complex sums looks very much like a "Gaussian distribution" (the famous Bell Curve). This allowed them to use well-known statistics to predict the outcome of these very strange, random number games.

Summary

In short, this paper confirms that a specific type of "super-cancellation" happens in random number sums, and it proves that this phenomenon is robust. It survives even when you twist the random numbers with the deep, complex mathematics of Modular Forms.

It tells us that in the chaotic world of number theory, there is a hidden, quiet order that refuses to be drowned out, no matter how complex the background noise gets.

Technical Summary

1. Problem Statement

The paper addresses the problem of determining the order of magnitude of the low moments ($0 \le q \le 1$) of sums of the form:
$$S(x) = \sum_{n \le x} h(n)\lambda(n)$$
where:

• $h(n)$ is a random multiplicative function, specifically either a Steinhaus random multiplicative function (values on the unit circle) or a Rademacher random multiplicative function (values $\pm 1$ on square-free integers).
• $\lambda(n)$ represents the Fourier coefficients of a fixed holomorphic Hecke eigenform $f$ of weight $\kappa \equiv 0 \pmod 4$ for the full modular group $SL_2(\mathbb{Z})$.

Context:
Standard heuristics (square-root cancellation) suggest that such sums should be of size roughly $\sqrt{x}$. However, A. J. Harper previously showed that for pure random multiplicative functions (without the twist $\lambda(n)$), the low moments are smaller than the square-root bound due to "large deviations" where the sum is unusually small. The authors investigate whether the arithmetic structure of the modular form coefficients $\lambda(n)$ alters this behavior or if the Harper phenomenon persists.

2. Methodology

The authors employ a sophisticated probabilistic and analytic number theory framework, largely extending the techniques introduced by A. J. Harper in his seminal work on random multiplicative functions. The methodology involves:

• Euler Product Approximation: The sum $S(x)$ is approximated by partial Euler products $F_{k,f}(s)$ over $P$-smooth numbers (numbers with prime factors $\le P$). The sum is decomposed based on the size of the largest prime factor of $n$.
• Change of Measure (Girsanov-type transformations): The authors define new probability measures ($\tilde{P}$) weighted by the Euler products. This allows them to transform the problem of estimating moments into estimating the probability that certain random walks (logarithms of the Euler products) stay within specific bounds.
• Gaussian Approximation: Using Berry-Esseen inequalities and properties of the coefficients $\lambda(n)$, the authors show that the logarithms of the Euler products behave asymptotically like sums of independent Gaussian random variables.
• Arithmetic Properties of $\lambda(n)$: A crucial part of the proof involves utilizing specific properties of the modular form coefficients:
  • Multiplicativity and the relation to the symmetric square $L$-function $L(s, \text{sym}^2 f)$.
  • Deligne's bound $|\lambda(n)| \le d(n)$.
  • The asymptotic formula for $\sum_{n \le x} \lambda^2(n)$ (Rankin-Selberg method).
  • Estimates for sums over primes involving $\lambda^2(p)$ (e.g., $\sum_{p \le x} \frac{\lambda^2(p)}{p} \sim \log \log x$).
• Upper and Lower Bounds:
  • Upper Bounds: Derived using Hölder's inequality and the decomposition of the sum into smooth and non-smooth parts, reducing the problem to bounding integrals of the Euler products.
  • Lower Bounds: Derived using a "reverse" argument involving conditional expectations and the Khintchine-type inequality, showing that the sum is large on a set of significant probability.

3. Key Contributions

1. Extension of Harper's Results: The paper successfully extends Harper's results on the low moments of random multiplicative functions to the case where the function is twisted by Fourier coefficients of modular forms.
2. Handling Arithmetic Complexity: Unlike the case of pure random multiplicative functions, the coefficients $\lambda(n)$ are not independent random variables; they are deterministic arithmetic functions with specific correlations. The authors demonstrate that despite these correlations, the "random" behavior of $h(n)$ dominates, and the arithmetic structure of $\lambda(n)$ (specifically its connection to the symmetric square $L$-function) does not change the order of magnitude of the moments.
3. Unified Treatment: The paper provides a unified proof for both Steinhaus and Rademacher cases, adapting the probabilistic machinery (change of measure and Gaussian approximation) to handle the specific algebraic relations of $\lambda(n)$ (e.g., $\lambda(p^m)$ relations and the real-valued nature of $\lambda(n)$).

4. Main Results

The central theorem (Theorem 1.1) establishes that for both Steinhaus and Rademacher random multiplicative functions $h(n)$, and for any $0 \le q \le 1$, the low moments satisfy:
$$\mathbb{E}\left| \sum_{n \le x} h(n)\lambda(n) \right|^{2q} \asymp \left( \frac{x}{1 + (1-q)\sqrt{\log \log x}} \right)^q$$
Interpretation:

• The order of magnitude is identical to the case where $\lambda(n) \equiv 1$ (the pure random multiplicative function case studied by Harper).
• The factor $1 + (1-q)\sqrt{\log \log x}$ in the denominator indicates that the sum is typically smaller than the trivial bound $x^q$ (which would correspond to $q=1$) and significantly smaller than the square-root heuristic $x^{1/2}$ when $q=1/2$.
• Specifically, for $q=1/2$, the expected magnitude is $o(\sqrt{x})$, confirming that the "square-root cancellation" heuristic is an overestimate for these sums.

5. Significance

• Robustness of Random Models: The result confirms that the "large deviation" behavior observed in random multiplicative functions is robust even when twisted by non-trivial arithmetic sequences like modular form coefficients. This suggests that the random multiplicative function model is a powerful tool for predicting the behavior of sums involving complex arithmetic functions.
• Modular Forms and Probability: The paper bridges the gap between the analytic theory of modular forms (specifically the distribution of Fourier coefficients) and probabilistic number theory. It shows that the specific arithmetic constraints of $\lambda(n)$ (governed by the Ramanujan-Petersson conjecture/Deligne's theorem) do not disrupt the probabilistic cancellation mechanisms driven by the random signs/phases of $h(n)$.
• Methodological Advancement: The techniques developed, particularly the adaptation of the Girsanov change of measure to handle the specific correlations in $\lambda(n)$ (via the symmetric square $L$-function), provide a template for analyzing other twisted sums involving automorphic forms.

In conclusion, Gao and Zhao prove that the "low moment" phenomenon is a universal feature of random multiplicative functions, persisting even in the presence of the rich arithmetic structure of modular forms.