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Oscillations of random multiplicative functions under initial bias

This paper resolves a conjecture by Kucheriaviy and a problem posed by Aymone by proving that random completely multiplicative functions, even when biased to be positive on small primes, exhibit oscillating partial sums that eventually become negative and change signs infinitely often.

Original authors: Rodrigo Angelo, Max Wenqiang Xu

Published 2026-03-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Rodrigo Angelo, Max Wenqiang Xu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a very long, chaotic game of "Heads or Tails," but with a twist.

The Game: A Random Multiplicative Function

In this paper, the authors are studying a mathematical object called a random completely multiplicative function. Let's call it "The Random Walker."

The Rules: The Walker moves along the number line (1, 2, 3, 4...). At every prime number (2, 3, 5, 7...), the Walker flips a coin.
- If it's Heads, the Walker steps Forward (+1).
- If it's Tails, the Walker steps Backward (-1).
The Multiplication: Once the Walker decides the direction for a prime, it sticks to that rule for all multiples of that prime. For example, if the Walker decides 2 is "Forward," then 4, 6, 8, 10, etc., are all influenced by that "Forward" momentum.
The Goal: We want to know: Does the Walker ever get stuck on one side of the starting line? In other words, if we add up all the steps the Walker has taken so far, will the total sum stay positive (or stay negative) forever, or will it eventually cross back and forth?

The Two Big Questions

The paper solves two specific puzzles about this Walker.

Puzzle 1: The "Head Start" Bias

Imagine someone cheats at the beginning of the game. They force the Walker to take only "Forward" steps for all the small primes (up to a certain point, say, up to the square of the logarithm of the total distance).

The Intuition: If you give the Walker a huge head start of "Forward" steps, surely it will stay positive for a long time, right?
The Old Guess: A mathematician named Kucheriaviy guessed that if you give the Walker a massive head start (covering primes up to a specific, very large threshold), it might stay positive forever.
The New Discovery: The authors of this paper say: "Nope."
- The Analogy: Imagine you are pushing a heavy shopping cart up a hill. You give it a massive, super-strong shove at the bottom (the "initial bias"). You might think it will roll all the way to the top without rolling back. But the hill is so bumpy (the randomness of the later primes) and the cart is so heavy that eventually, the random bumps will knock it backward.
- The Result: Even with a huge head start, the probability that the Walker stays positive for the entire journey is effectively zero. The randomness of the later numbers is too strong; the Walker will eventually swing back and cross the zero line.

Puzzle 2: The "Infinite Dance" (The Square Root Problem)

The second puzzle is about a slightly different version of the game. Instead of just adding the steps, we weigh them. We divide the step size by the square root of the number ( $\sqrt{n}$ ). This makes the steps get smaller and smaller as the game goes on.

The Question: If we let this game go on forever, will the Walker eventually stop changing direction? Will it decide, "Okay, I'm going to stay positive from now on," or "I'm going to stay negative"?
The Context:
- If we weigh the steps heavily (divide by a large number), the Walker settles down quickly and stops changing signs.
- If we weigh them lightly, the Walker goes crazy and changes signs constantly.
- The "Square Root" case is the tipping point. It's the exact middle ground where mathematicians weren't sure what would happen.
The Discovery: The authors prove that even at this tipping point, the Walker never settles down.
- The Analogy: Imagine a drunk person walking on a tightrope. If the rope is very stiff, they might eventually find their balance and stop wobbling. If the rope is very loose, they fall off immediately. But here, the rope is just right. The authors prove that this drunk person will never stop wobbling. They will cross the center line (change signs) an infinite number of times. They will dance back and forth forever, never finding a permanent resting spot.

How Did They Solve It? (The Secret Sauce)

The authors didn't just guess; they used a clever mathematical trick involving time and speed.

The "Speedometer" Trick: They looked at the Walker's behavior not just at one moment, but at different "speeds" (mathematically, different values of $t$ ).
The Expectation: If the Walker stayed positive forever, its "speed" would have to slow down in a very specific, predictable way (like a car coasting to a stop).
The Reality: They proved that the Walker's speed actually behaves like a random rollercoaster. It goes up and down wildly.
The Conclusion: Because the "speed" is so chaotic and unpredictable, it is statistically impossible for the Walker to maintain a smooth, one-sided path. The chaos of the random coin flips eventually forces the Walker to change direction.

Why Does This Matter?

This isn't just about a math game. These "Random Walkers" are models for some of the most mysterious numbers in mathematics, like the Riemann Zeta function (which is related to the distribution of prime numbers).

Understanding how these random sums behave helps mathematicians understand the "noise" in the universe of numbers.
It confirms that randomness is incredibly powerful. Even if you rig the game at the start, the long-term chaos of the system will eventually overwhelm your setup.
It solves a decades-old debate about whether these mathematical objects ever "settle down" or if they are destined to oscillate forever.

In short: The paper proves that in the chaotic world of random numbers, you can't cheat your way to a permanent win. The system is too wild, and it will always swing back and forth, forever.

1. Problem Statement

The paper addresses two fundamental problems regarding the sign changes of partial sums of a random completely multiplicative function $f(n)$ . This function is defined by setting $f(p) = \pm 1$ with probability $1/2$ independently for each prime $p$ , and extending it multiplicatively to all integers $n$ .

The two specific problems solved are:

The Effect of Initial Bias: How does fixing the values of $f(p)$ for small primes (an "initial bias") affect the probability that the partial sums $S(N) = \sum_{n \le N} f(n)$ remain non-negative for all $N \le x$ ? This resolves a conjecture by Kucheriaviy regarding the threshold of the bias length $y$ .
The Critical Case of Weighted Sums: Do the partial sums of the weighted series $\sum \frac{f(n)}{\sqrt{n}}$ change signs infinitely often? Previous work had settled the cases for weights $n^{-\sigma}$ where $\sigma > 1/2$ (finite sign changes) and $\sigma < 1/2$ (infinite sign changes), leaving the critical case $\sigma = 1/2$ open (a problem posed by Aymone).

2. Methodology

The authors develop a novel approach that avoids the standard technique of analyzing the function on complex lines near the critical line (which relies on constraints on the maximum modulus of Dirichlet series). Instead, they utilize the monotonicity properties of the series for real-valued $s$ derived from the non-negativity of partial sums.

Key Technical Tools:

Euler Product Approximation: The authors relate the partial sums to the Euler product of the Dirichlet series $F(s) = \sum f(n)n^{-s}$ . For $s = 1/2 + t$ , this is approximated by $\exp(R(t))$ , where $R(t)$ is a sum over primes.
Monotonicity Constraint (Lemma 2.3 & 3.1): They prove that if the partial sums are non-negative, the function $F(s)/s$ (or related weighted sums) must exhibit approximately decreasing behavior as $s$ increases (or as $t$ decreases).
Central Limit Theorem (CLT) for Differences: The core of the argument involves analyzing the differences $R(t) - R(2t)$ $R (t) - R (2 t)$ .
- $R(t)$ behaves like the logarithm of the series.
- $R(t) - R(2t)$ behaves like a discrete derivative.
- The authors prove that for a sequence of $t_i$ growing geometrically, the vector of differences $(R(t_i) - R(2t_i))$ converges in distribution to a joint normal distribution with independent components.
Contradiction via Oscillation: The monotonicity required by non-negative partial sums implies that $R(t) - R(2t)$ must be bounded above by a constant. However, the CLT shows that these differences behave like independent Gaussian variables with a positive mean ( $\log 2 / 2$ ). The probability that all such independent variables stay below a fixed bound simultaneously decays to zero as the number of variables increases.

3. Key Contributions and Results

Theorem 1.1: Sharpness of the Initial Bias Threshold

Context: Kucheriaviy previously showed that if $f(p)=1$ for all $p \le y$ with $y \ge C \frac{(\log x)^2 \log \log \log x}{\log \log x}$ , the probability of non-negative partial sums up to $x$ is $1 - o(1)$ .
Result: The authors prove the complementary result. If $y = o\left( \left(\frac{\log x}{\log \log x}\right)^2 \right)$ , then conditional on $f(p)=1$ for $p \le y$ , the probability that $\sum_{n \le N} f(n) \ge 0$ for all $N \le x$ is $o(1)$ as $x \to \infty$ .
Significance: This confirms Kucheriaviy's conjecture that the threshold $y \approx (\log x)^2$ is sharp. Below this scale, the "randomness" of the remaining primes is sufficient to force sign changes, regardless of the initial bias.

Theorem 1.2: Infinite Sign Changes at $\sigma = 1/2$

Context: For $\sigma > 1/2$ , the series converges to a positive Euler product (finite sign changes). For $\sigma < 1/2$ , Aymone proved infinite sign changes. The case $\sigma = 1/2$ was unresolved.
Result: The authors prove that almost surely, the partial sums of $\sum_{n \le N} \frac{f(n)}{\sqrt{n}}$ change signs infinitely many times.
Significance: This closes the gap in the understanding of the critical exponent for random multiplicative functions. It demonstrates that even at the critical boundary where the series is "barely" convergent, the random fluctuations are strong enough to prevent the sum from settling into a single sign.

4. Significance and Implications

Resolution of Open Problems: The paper definitively solves two conjectures/problems (Kucheriaviy's and Aymone's) that were central to the study of random multiplicative functions.
Methodological Shift: The paper introduces a powerful technique that relies on real-variable analysis (monotonicity of Dirichlet series) combined with probabilistic limit theorems (CLT on prime sums). This bypasses the need for complex analysis techniques (like bounding maximums on vertical lines) which were previously the standard tool for such problems but were insufficient for these specific cases.
Understanding of Bias: The results clarify the interplay between "initial conditions" (fixed small primes) and "random noise" (large primes). It shows that the variance contributed by primes larger than $y$ dominates the behavior of the sum unless $y$ is extremely large (specifically, near $(\log x)^2$ ).
Connection to Other Fields: The authors note connections to the Fyodorov-Hiary-Keating conjecture and the study of quadratic characters, suggesting their methodology could be applicable to other problems involving oscillations of arithmetic functions.

In summary, the paper establishes that random completely multiplicative functions are inherently oscillatory. Even with a significant initial bias or at the critical weighting of $1/\sqrt{n}$ , the cumulative effect of the random signs of larger primes forces the partial sums to change signs infinitely often or with probability tending to zero.