REVIEW
A Goldbach-Type Problem for the Unitary M\"obius Function
Number Theory · Submitted 7 Nov 2026 · 25 references
Pith's verdict: Solid, well-executed paper on additive patterns of the unitary Möbius function; the results are new, the proofs are clear, and the computer verification is thorough.
Why Pith says this
Highlight any text to comment on it.
Referee report
The paper
The paper claims
Every sufficiently large integer yields all four sign patterns of the unitary Möbius function at least N/(128 (log N)^3) times, and for N\ge18 at least once.
The reading
What carries the argument
The key machinery is a set of four additive decompositions of $N$ parameterized by an odd prime $q$ satisfying $q\nmid N$ and every odd prime divisor of $q+1$ dividing $N$ (the relation $H(N,q)$). Lemma 3.2 shows that the product of the eight $g$-values from these decompositions is $-1$, forcing both positive and negative correlation products. For the quantitative lower bound, uniform bounds for the exponential sum $G_x(\alpha)=\sum_{n\le x}g(n)e(\alpha n)$ (Proposition 3.7) and a mean-square estimate for $C_g(N)=\sum_{a<N}g(a)g(N-a)$ (Proposition 3.8) are used. For the explicit existence threshold, a covering-chain construction (Lemma 5.3) allows finite data to handle all large integers, supplemented by exhaustive verification for $N<100000$. The tail case $q(N)\ge 61$ uses a uniform interval-length condition (Proposition 5.8) and Nagura's theorem.
Extended reading notes
Core claim
The central discovery is that the sign sets $A_s=\{n:g(n)=s\}$ ($s\in\{-1,+1\}$) satisfy a Goldbach-type property: every sufficiently large integer belongs to all four sumsets $A_s+A_t$. The lower bound for each pattern is $N/(128(\log N)^3)$, proved via a construction involving the least odd prime not dividing $N$ and exponential-sum estimates. A separate finite-verification argument shows that for $N\ge 18$ all four patterns occur simultaneously, and the optimal thresholds are $4$ (for negative-negative), $8$ (for mixed), and $18$ (for positive-positive). Almost all integers satisfy the asymptotic $R_{st}(N)=N/4+o(N)$.
Load-bearing premise
The weakest assumption is the correctness of the finite verification software and data for $N<100000$ and for the covering chains; the authors acknowledge this by providing an independent verifier and hash manifests.
Editorial extensions
If this is right
- Every sufficiently large integer belongs to all four sumsets $A_s+A_t$ where $A_s=\{n:g(n)=s\}$.
- The thresholds 4, 8, 18 are optimal for the individual patterns, and 18 is the common threshold for all four patterns to occur simultaneously.
- For almost all $N$, the representation count is asymptotically $N/4$ for each pattern, with a logarithmic saving.
- The proof provides a template for Goldbach-type results with arithmetic functions whose sign sets have density $1/2$ and satisfy a scaling property similar to (2.2).
- The corresponding problem for the ordinary Möbius function remains open, reduced to proving $T(N)=\sum_{a<N}\mu(a)\mu(N-a)=o(N)$.
Reading between the lines
- The lower bound $N/(\log N)^3$ is likely far from optimal; a natural conjecture is that $R_{st}(N)\sim N/4$ for all $N$.
- The covering-chain method could be adapted to other multiplicative functions with positive density and a simple scaling rule, such as the Liouville function's level sets.
- The finite verification up to 100000, while rigorous, suggests that a purely analytic proof of the threshold 18 might be possible with more refined exponential-sum estimates.
Original abstract
Let $g(n)=(-1)^{\omega(n)}$, where $\omega(n)$ denotes the number of distinct prime factors of $n$. We prove that, for every sufficiently large integer $N$, each ordered additive sign pattern $(g(a),g(N-a))=(s,t)$ occurs at least $N/(128(\log N)^3)$ times. We also prove that, for every $N\ge18$ and every $s,t\in\sgnset$, there is at least one integer $1\le a<N$ such that $(g(a),g(N-a))=(s,t)$. In addition, we establish a one-quarter asymptotic formula for almost all target integers. Finally, we discuss Goldbach-type problems for the ordinary M\"obius function and formulate related questions.
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
Let $g(n)=(-1)^{\omega(n)}$, where $\omega(n)$ denotes the number of distinct prime factors of $n$. We prove that, for every sufficiently large integer $N$, each ordered additive sign pattern $(g(a),g(N-a))=(s,t)$ occurs at least $N/(128(\log N)^3)$ times. We also prove that, for every $N\ge18$ and every $s,t\in\sgnset$, there is at least one integer $1\le a<N$ such that $(g(a),g(N-a))=(s,t)$. In addition, we establish a one-quarter asymptotic formula for almost all target integers. Finally, we discuss Goldbach-type problems for the ordinary M\"obius function and formulate related questions.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $g(n)=(-1)^{\omega(n)}$, where $\omega(n)$ denotes the number of distinct prime factors of $n$. | background 0.96 | 0.18 | 0.02 | asserted 0.55 |
| s1 | We prove that, for every sufficiently large integer $N$, each ordered additive sign pattern $(g(a),g(N-a))=(s,t)$ occurs at least $N/(128(\log N)^3)$ times. | result 1.00 | 0.97 | 0.69 | asserted 1.00 |
| s2 | We also prove that, for every $N\ge18$ and every $s,t\in\sgnset$, there is at least one integer $1\le a<N$ such that $(g(a),g(N-a))=(s,t)$. | result 0.99 | 0.96 | 0.18 | asserted 1.00 |
| s3 | In addition, we establish a one-quarter asymptotic formula for almost all target integers. | result 1.00 | 0.95 | 0.12 | asserted 1.00 |
| s4 | Finally, we discuss Goldbach-type problems for the ordinary M\"obius function and formulate related questions. | future work 0.93 | 0.80 | 0.03 | asserted 0.58 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
References
Discussion
Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.