REVIEW
On a conjecture of Corradi and Katai
Number Theory · Submitted 13 Aug 2026 · 6 references
Pith's verdict: A clean, correct, and mildly novel note that unconditionally settles the liminf form of the Corrádi–Kátai conjecture via Davenport's uniform bound; the two displayed algebraic slips are repairable and do not affect the main theorems.
T0 · AI review
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Referee report
The paper
The paper claims
The paper proves that for every fixed k≥3, the k-fold Goldbach-type Liouville sums have cancellation of any fixed power of log N, that the binary sums satisfy liminf |G_2(N)|/N = 0, and that all 2^k sign patterns occur among k-term…
The reading
What carries the argument
The carrying object is the $L^2$-normalized Liouville exponential sum $S(N,x) = N^{-1/2}\sum_{n \le N} \lambda(n) e(nx)$. The identity that does the work is $\int_0^1 |S(N,x)|^4\,dx = N^{-2}\sum_{n=0}^N |G_2(n)|^2$, which converts pointwise control of $S(N,x)$ into a second-moment bound on the binary sums $G_2(n)$. Since the $L^2$ norm of $S(N,x)$ is $1$ and the uniform bound $|S(N,x)| \ll_A \sqrt{N}/\log^A N$ holds, the fourth moment is $O_A(N/\log^A N)$; partial summation turns this into Proposition 1, from which all three main theorems follow.
Extended reading notes
Core claim
The central claim of the paper is square-averaged cancellation for the binary sums: $\sum_{n \le N} |G_2(n)/n|^2 \ll_A N/\log^A N$. From this statement the paper derives the log-power bound for all $k \ge 3$ by a short induction, the density-zero statement for the exceptional set in Theorem 2, and the equidistribution of Liouville sign patterns in Theorem 3. The full limit $\lim_{N\to\infty} |G_2(N)|/N = 0$ is not claimed; what is proved is that any limit point of the sequence must be $0$, so the conjecture would follow if the ratio converges. The proof is not conditional on any unproved hypothesis once the classical uniform exponential-sum bound for $\lambda$ is accepted.
Load-bearing premise
The load-bearing premise is the classical uniform bound $|S(N,x)| \ll_A \sqrt{N}/\log^A N$ for the normalized Liouville exponential sum, quoted rather than proved here, since the fourth-moment estimate and every theorem that follows from it would fail if that bound were weaker.
Editorial extensions
If this is right
- For each fixed $k \ge 3$ and each $(\varepsilon_1,\dots,\varepsilon_k) \in \{\pm1\}^k$, the number of $k$-tuples $(a_1,\dots,a_k)$ with sum $N$ and $\lambda(a_i)=\varepsilon_i$ is $2^{-k}\binom{N-1}{k-1} + O_A(N^{k-1}/\log^A N)$.
- The exceptional set $K(\delta) = \{N : |G_2(N)| > \delta N\}$ has counting function $O_A(\delta^{-2} N/\log^A N)$, hence density zero; consequently $\liminf_N |G_2(N)|/N = 0$.
- If the ratio $|G_2(N)|/N$ has any limit as $N \to \infty$, that limit must be $0$, reducing the Corrádi–Kátai conjecture to proving convergence of the ratio.
- The proof shows Theorem 1 and Theorem 3 are equivalent: sign-pattern equidistribution and log-power cancellation of $G_k$ stand or fall together.
Reading between the lines
- Going beyond the paper, the same fourth-moment mechanism should give log-power cancellation for $k$-fold convolutions of any bounded arithmetic function whose normalized exponential sum satisfies a uniform $O(\sqrt{N}/\log^A N)$ bound; the proof never uses multiplicativity of $\lambda$ beyond boundedness.
- One testable extension is to replace $\lambda$ by the Möbius function or by a nontrivial Dirichlet character and check whether the analogue of Proposition 1, $\sum_{n\le N}|G_2(n)|^2/n^2 \ll N/\log^A N$, still holds; if it does, the equidistribution statements should transfer.
- The density-zero conclusion for $K(\delta)$ leaves open a stronger possibility the paper does not address: that $|G_2(N)|/N$ decays like a power of $1/\log N$ for all large $N$, consistent with the conditional lower bounds for the defect referenced in the introduction.
Original abstract
We consider Goldbach type sums corresponding to the Liouville function and prove the existence of sufficient cancellations. We also consider applications to sign patterns in the Liouville function.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We consider Goldbach type sums corresponding to the Liouville function and prove the existence of sufficient cancellations. We also consider applications to sign patterns in the Liouville function.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider Goldbach type sums corresponding to the Liouville function and prove the existence of sufficient cancellations. | contribution 0.63 | 0.95 | 0.77 | asserted 1.00 |
| s1 | We also consider applications to sign patterns in the Liouville function. | contribution 0.97 | 0.82 | 0.08 | unclear 0.60 |
What was examined, and what was not
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References
Discussion
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