On a conjecture of Corradi and Katai

Technical Summary: On a Conjecture of Corrádi and Kátai

Problem Statement
This paper investigates the asymptotic behavior of Goldbach-type sums involving the Liouville function, $\lambda(n)$. While the classical Goldbach conjecture concerns the representation of even integers as sums of two primes (related to the von Mangoldt function $\Lambda(n)$), this work replaces $\Lambda(n)$ with the completely multiplicative Liouville function, defined by $\lambda(p) = -1$ for all primes $p$ and $\lambda(mn) = \lambda(m)\lambda(n)$.

The central object of study is the sum:
$$G_k(N) := \sum_{(a_1, \dots, a_k) \in S_k(N)} \lambda(a_1) \cdots \lambda(a_k)$$
where $S_k(N) = \{(a_1, \dots, a_k) \in \mathbb{N}^k \mid a_1 + \dots + a_k = N\}$. The trivial bound for this sum is $|G_k(N)| \ll N^{k-1}$. The primary goal is to establish stronger bounds demonstrating significant cancellation in these sums, particularly for the case $k=2$, which relates to the Corrádi-Kátai conjecture.

Methodology
The author employs techniques from Fourier analysis and exponential sums, specifically leveraging results by Davenport regarding the Liouville function.

1. Fourier Analysis and Exponential Sums: The paper utilizes the exponential sum $S(N, x) = \frac{1}{\sqrt{N}} \sum_{n=1}^N \lambda(n)e(nx)$. The core analytic tool is Davenport's theorem, which states that $|S(N, x)| \ll_A \frac{\sqrt{N}}{\log^A N}$ uniformly for $x \in [0, 1]$.
2. Integral Representations: Using Parseval's identity and properties of exponential sums, the author relates the sums $G_k(N)$ to integrals of powers of $S(N, x)$. Specifically, the identity $\int_0^1 S^2(N, x)e(-nx)dx = \frac{G_2(n)}{N}$ is used to connect the discrete sums to the continuous $L^2$ and $L^4$ norms of the exponential sum.
3. Inductive Arguments: For the case $k \geq 3$, the proofs proceed by induction. The base case ($k=3$) relies on the Cauchy-Schwarz inequality applied to the $k=2$ case, which is bounded using the $L^2$ average of $G_2(n)$ derived from Davenport's theorem.
4. Partial Summation: To derive bounds for the set of $N$ where $|G_2(N)|$ is large, the author uses partial summation techniques on the mean square estimate of $G_2(n)/n$.

Key Contributions and Results

• Theorem 1 (Case $k \geq 3$): For any fixed integer $k \geq 3$ and any positive constant $A$, the paper proves:
  $$|G_k(N)| \ll_{A, k} \frac{N^{k-1}}{\log^A N}$$
  This result establishes a power-saving improvement over the trivial bound $O(N^{k-1})$ for all ranks $k \geq 3$.

• Theorem 2 (Case $k = 2$): The paper addresses the more difficult case of $k=2$. It defines $K(\delta) = \{N \in \mathbb{N} \mid |G_2(N)| > \delta N\}$. The main result here is a density estimate:
  $$|K(\delta) \cap \{1, 2, \dots, N\}| \ll_A \frac{1}{\delta^2} \frac{N}{\log^A N}$$
  This implies that the set of integers where $|G_2(N)|$ exceeds a linear fraction of $N$ has density zero. Consequently, $\liminf_{N \to \infty} \frac{|G_2(N)|}{N} = 0$. The paper notes that this result asserts the Corrádi-Kátai conjecture is true should the limit defining the conjecture exist, but does not prove the limit exists unconditionally.

• Theorem 3 (Sign Distribution): As a corollary of the main results, the paper demonstrates the equi-distribution of sign patterns in the Liouville function for $k \geq 3$. For any sequence of signs $(\epsilon_1, \dots, \epsilon_k) \in \{\pm 1\}^k$, the number of solutions to $a_1 + \dots + a_k = N$ with $\lambda(a_i) = \epsilon_i$ is:
  $$\frac{1}{2^k} \binom{N-1}{k-1} + O_A\left(\frac{N^{k-1}}{\log^A N}\right)$$
  This indicates that sign patterns occur with the expected frequency up to a small error term.

Significance and Context
The paper situates its results within the context of the Corrádi-Kátai Conjecture (1969), which posits that $\lim_{N \to \infty} \frac{|G_2(N)|}{N} = 0$.

• The author notes that while the conjecture was proven conditionally (assuming the existence of infinitely many Siegel zeros) in a cited work [3], an unconditional proof has remained elusive.
• The paper references recent work by Mangerel, who proved $|G_2(N)| < N-1$ for $N \notin \{2, 3, 5, 10\}$, answering a weaker question posed by Sarnak.
• The author's contribution is modest regarding the full conjecture: Theorem 2 does not prove the limit exists or is zero for every $N$, but it proves that the limit inferior is zero and that large values of $|G_2(N)|$ are extremely sparse.
• The paper asserts that Theorem 1 and Theorem 3 are equivalent, highlighting that the cancellation in the sums is directly linked to the random-like distribution of the Liouville function's signs.

The work provides a rigorous unconditional improvement over trivial bounds for higher-rank sums ($k \geq 3$) and offers a partial, density-based resolution to the long-standing $k=2$ case, reinforcing the heuristic that the Liouville function behaves randomly in additive settings.