🔢 mathematics

On variants of Chowla's conjecture

This paper provides combinatorial proofs for two recent results related to Chowla's conjecture by analyzing shifted convolution sums of completely multiplicative functions with values in $\{\pm 1\}$ and determining their corresponding spectrum.

Original authors: Krishnarjun Krishnamoorthy

Published 2026-08-13

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Krishnarjun Krishnamoorthy

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 the number line as an endless highway stretching into the horizon, where every integer is a stop on the route. Some of these stops are special "prime" stations, the building blocks of all other numbers. Now, imagine a mysterious game played with these numbers, where we assign them a secret code: either a plus sign (+1) or a minus sign (-1). The rules for assigning these codes are strict and follow a pattern called "multiplicativity," meaning the code for a big number is just the product of the codes for its smaller prime parts.

The big question mathematicians have been asking for decades is: if you look at a long stretch of this highway, do these plus and minus signs cancel each other out perfectly? Or do they clump together in weird patterns? This is the heart of "Chowla's Conjecture." It suggests that the signs should be completely random, like flipping a fair coin over and over again, so that if you average them out over a long distance, the result should be zero. If they don't cancel out, it would mean there's a hidden order or a secret connection between numbers that are close to each other, which would shake up our understanding of how numbers work.

The Paper's Discovery

In this paper, Krishnarjun Krishnamoorthy steps into this game to investigate two specific scenarios involving these plus-and-minus codes. The author isn't just guessing; they provide a "combinatorial proof," which is like solving a puzzle by rearranging the pieces logically rather than using heavy, complex machinery.

First, the author looks at what happens when we pick a very specific, "small" collection of prime numbers to be the ones that get the minus sign (-1), while all other primes get a plus sign (+1). They ask: if we shift our view down the highway by a few steps (looking at $n$ , $n+1$ , $n+2$ , etc., all at once), what is the average of the product of their signs?

The paper proves that for these "small" sets of primes, the average doesn't just vanish into nothingness; it settles on a specific, predictable number. This number is calculated by multiplying together a tiny fraction for every single prime in that small set. It's like saying the final score of the game is the result of a chain reaction of tiny adjustments, one for each prime you chose. The author shows that even though the pattern of signs might look messy, the long-term average follows a neat formula that looks like a product of local rules.

Second, the paper tackles a more dramatic question: Can the average ever be perfect? Could the signs align so perfectly that the average is exactly +1 or exactly -1? The paper proves that the answer is a hard "no," unless you choose no primes at all (which is the boring, empty case where everything is just +1). If you pick even one prime to be a minus, the average will never reach the extreme limits of +1 or -1. It will always be slightly less than perfect, proving that there is always some "noise" or randomness in the system. This confirms a recent, more complex result using a simpler, more direct method.

Finally, the author maps out the "spectrum" of possible outcomes. They show that by carefully choosing different small sets of primes, you can hit almost any number between 0 and 1 (and even some negative numbers) as your average. It's as if the author has built a dial that can be tuned to produce a wide variety of specific averages, proving that the behavior of these numbers is rich and varied, but always bound by the rules they discovered.

In short, the paper doesn't solve the entire mystery of the number highway, but it builds a solid bridge across a tricky section. It proves that for certain types of rules, the average behavior is predictable and calculable, and it definitively rules out the possibility of the signs ever becoming perfectly uniform if any variation is introduced.

Technical Summary: On Variants of Chowla's Conjecture

Problem Statement
This paper investigates the shifted convolution sums associated with completely multiplicative functions $f: \mathbb{N} \to \{\pm 1\}$ . The central focus is the behavior of the average:
$\lim_{x \to \infty} \frac{1}{x} \sum_{n \le x} f(n+h_1) \cdots f(n+h_d)$
where $H = \{h_1, \dots, h_d\}$ is a fixed set of non-negative integers. This problem is a direct variant of Chowla's Conjecture, which posits that for the Liouville function $\lambda$ (where $f(p)=-1$ for all primes), this limit vanishes for any non-empty $H$ . The paper specifically examines functions of the form $\lambda_P$ , defined by a subset of primes $P$ such that $\lambda_P(p) = -1$ if $p \in P$ and $1$ otherwise. The study addresses two main questions: the evaluation of these limits when $P$ is a "small set" (a set of primes with a convergent sum of reciprocals) and the determination of the "spectrum" of possible limit values.

Methodology
The author employs elementary combinatorial arguments and structural analysis of subsets of natural numbers, avoiding heavy analytic machinery such as Tauberian theorems or deep ergodic theory, which are common in recent literature on this topic.

Key methodological components include:

Symmetric Difference Structure: The paper utilizes the group isomorphism between subsets of primes (under symmetric difference) and completely multiplicative functions (under pointwise multiplication). This allows the decomposition of complex correlation sums into simpler components via Lemma 2, which relates the sets $N^H_P = \{n \mid \Lambda^H_P(n) = -1\}$ under changes to $P$ and $H$ .
Density Approximation: The proofs rely on approximating the natural density of the set $N^H_P$ using finite unions of arithmetic progressions ( $C(r, s)$ ). By constructing upper and lower bounds ( $X^H_P(r)$ and $Y^H_P(r)$ ) within the algebra generated by these progressions, the author establishes the existence of the natural density $\eta^H_P$ .
Inductive Limiting Processes: For finite sets of primes, the density is calculated explicitly. For infinite "small" sets, the author uses a limiting process, showing that the error term in the density approximation is bounded by the sum of reciprocals of the remaining primes. This ensures convergence only when $P$ is a small set.
Combinatorial Reduction for Theorem 2: To address the case where the limit might equal $\pm 1$ , the paper employs a combinatorial argument (Lemma 5, attributed to Wildon) regarding families of sets closed under symmetric difference and translation. This reduces the problem to the two-point correlation case, which is settled by invoking the Matomäki–Radziwiłł theorem.

Key Contributions and Results

Theorem 1 (Small Sets of Primes): For a fixed $H$ and a small set of primes $P$ , the limit $\kappa^H_P = \lim_{x \to \infty} \frac{1}{x} \sum_{n \le x} \Lambda^H_P(n)$ exists and is given by an Euler-like product:
$\kappa^H_P = \prod_{p \in P} (1 - 2\eta^H_p)$
where $\eta^H_p$ is the natural density of integers $n$ such that $\Lambda^H_{\{p\}}(n) = -1$ . For non-exceptional primes (those not dividing differences in $H$ ), $\eta^H_p = \frac{d}{p+1}$ . The author notes that while this result can be deduced from prior work (specifically [10]), the provided proof is elementary and combinatorial.
Theorem 2 (Non-Constant Functions): If $\lambda_P$ is a non-constant completely multiplicative function (i.e., $P \neq \emptyset$ ), then the correlation average cannot be constantly $1$ or $-1$. Specifically:
$\liminf_{x \to \infty} \frac{1}{x} \sum_{n \le x} \Lambda^H_P(n) < 1 \quad \text{and} \quad \limsup_{x \to \infty} \frac{1}{x} \sum_{n \le x} \Lambda^H_P(n) > -1$
This result implies that for any non-trivial $P$ , the values are not perfectly correlated. The proof relies on the Matomäki–Radziwiłł theorem regarding short averages, but the paper provides a new combinatorial framework to apply it to this specific context.
Spectrum Determination: The paper defines the "spectrum" $\Gamma_H$ as the topological closure of the set of all possible values $\kappa^H_P$ as $P$ ranges over all small subsets of primes. The author proves that $\Gamma_H = [\alpha_H, 1] \cup [0, 1]$ , where $\alpha_H = \inf_p (1 - 2\eta^H_p)$ . This characterizes the full range of possible correlation values for small sets of primes.

Significance and Claims
The paper claims to provide combinatorial proofs for two recent results in the direction of Chowla's Conjecture, offering an alternative to the more complex analytic or ergodic-theoretic approaches found in the literature (such as those by Matomäki, Radziwiłł, and Teräväinen).

The author emphasizes that:

The derivation of the product formula in Theorem 1 is "comparatively elementary," despite the fact that the function $\Lambda^H_P$ is not multiplicative and the associated Dirichlet series may not extend beyond $\Re(s) > 1$ .
The multiplicative structure of the limit (the Euler product) aligns with the principle that global density is a product of local densities, even in the absence of multiplicativity in the underlying function.
The work clarifies the behavior of these sums specifically for "small" sets of primes, a case where the limit is non-zero, and contrasts this with the expectation that the limit vanishes for large sets (which would imply Chowla's Conjecture).

The paper does not claim to prove Chowla's Conjecture for the Liouville function (where $P$ is the set of all primes, a large set), noting that such a generalization is "out of reach" with the current methods. Instead, it serves to rigorously establish the behavior of these correlations for small perturbations of the Liouville function and to map the spectrum of possible values.

More like this