REVIEW
Density and distribution of quotients associated with Goldring's $w$-function
Number Theory · Submitted 2 Nov 2026 · 12 references
Pith's verdict: A correct resolution of the three Chen-Shi-Wu open problems on Goldring's w-function; the one fragile-looking step, the hand-checked Green-Tao complexity table, has been verified twice and holds up.
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Referee report
The paper
The paper claims
This paper proves that on products of three primes the quotients $R(n) = P(w(n))/P(n)$ and $S(n) = w(n)/n$ have all of $[0,1]$ and $[0,\infty)$ as limit-point sets, while $R(n)$ simultaneously converges to zero in relative density…
The reading
What carries the argument
The engine is a reduction of a target quotient value to simultaneous primality of a small system of linear forms, plus the unconditional complexity-at-most-two case of the Green-Tao theorem on linear equations in primes, which counts points where fixed homogeneous linear forms all take prime values; the complexity condition is a mild algebraic-span requirement on the forms' coefficient vectors. On $B_3$, imposing $p + q = 2mb$ forces $P(p+q) = b$, so $w(p^2q) = pb^2$ and both quotients become continuous, scale-invariant functions of $(a,b)$; Lemma 4 verifies local solubility and complexity at most one for the three forms $a$, $b$, $2mb - a$, giving $\gg X^{2/3}(\log X)^{-3}$ integers in any prescribed quotient interval. On $C_3$, imposing $p + q = 2mx$, $p + r = 2my$, $q + r = 2z$ makes $w(pqr) = xyz$, and Lemma 5 verifies local solubility and complexity at most two for the six forms $x, y, z, mx + my - z, mx - my + z, -mx + my + z$, giving $\gg X(\log X)^{-6}$ integers. The relative-density result runs in the opposite direction: $R(n) > \alpha$ forces $p + r$ or $q + r$ to be a small multiple of a prime exceeding $\alpha r$, and Lemma 6 shows such adjacent-prime coincidences are $o(T/\log T)$ uniformly in the shift, so they cannot occur for a positive proportion of $n$, with a fixed-shift Selberg sieve playing the same role on $B_3$. For the joint set, the identity $w(p^2q) = pP(p+q)^2$ collapses the two quotients on $B_3$ onto the one-parameter families $\Gamma_m$, and the closure of their union with the two coordinate axes is shown to contain exactly the finite joint limits.
Extended reading notes
Core claim
On its own terms, the paper's central claim is that the two quotients attached to Goldring's $w$-function are simultaneously as rich and as sparse as possible on each of the classes $B_3$ and $C_3$: the finite limit-point sets of $R(n)$ and $S(n)$ are exactly $[0,1]$ and $[0,\infty)$, and every nonempty open interval $I \subset (0,1)$ or $J \subset (0,\infty)$ contains values with counts $\gg_I X^{2/3}(\log X)^{-3}$ on $B_3$ and $\gg_I X(\log X)^{-6}$ on $C_3$ (with $w(n) \in C_3$ also imposed). At the same time, for every fixed $\alpha > 0$ the set of $n \le X$ lying in $A_3$, $B_3$ or $C_3$ with $R(n) > \alpha$ has size $o(S(X))$, so $D_{A_3}(\alpha) = 1$ for all $0 < \alpha \le 1$. On $B_3$ the paper describes the joint limit set of $(R(n), S(n))$ exactly as $([0,1] \times \{0\}) \cup (\{0\} \times [0,\infty)) \cup \bigcup_{m \ge 1} \Gamma_m$, where $\Gamma_m$ is the curve traced by $(r_m(u), g_m(u))$ for $u > 0$, with $r_m(u) = \max\{u, (u+1)/(2m)\}/\max\{u,1\}$ and $g_m(u) = (u+1)^2/(4m^2u)$; in particular the section at first coordinate $1/2$ is $\{(1/2,0)\} \cup \{(1/2, 9/(8m^2)) : m \ge 2\}$, so $(1/2, 1)$ is not a joint limit even though $1/2$ and $1$ are limits of the separate quotients.
Load-bearing premise
The load-bearing premise is the hand-checked verification in Lemma 5 that for every positive integer $m$ the six forms $x, y, z, mx + my - z, mx - my + z, -mx + my + z$ have complexity at most two and are locally soluble modulo every prime, since a single failure for some $m$ would break the squarefree interval-counting construction on $C_3$.
Editorial extensions
If this is right
- Problems 1 and 2 of Chen, Shi and Wu are answered affirmatively: $R(A_3)$ is dense in $[0,1]$ and $S(A_3)$ is dense in $[0,\infty)$, with quantitative counts $\gg$ in every open interval on both $B_3$ and $C_3$.
- Problem 3 is answered: $D_{A_3}(\alpha) = 1$ for every $0 < \alpha \le 1$, and more strongly the number of $n \le X$ in each of $A_3$, $B_3$, $C_3$ with $R(n) > \alpha$ is $o(S(X))$.
- Limit-point density and relative frequency point in opposite directions: every $\theta \in [0,1]$ is attained as a limit along sequences $n \to \infty$ in $B_3$ and $C_3$, yet the typical element of each class has $R(n)$ close to zero.
- The joint limit set on $B_3$ is exactly $([0,1] \times \{0\}) \cup (\{0\} \times [0,\infty)) \cup \bigcup_{m \ge 1} \Gamma_m$, and its section at first coordinate $1/2$ is $\{(1/2,0)\} \cup \{(1/2, 9/(8m^2)) : m \ge 2\}$, so knowing the separate limit sets of the two quotients does not determine their joint behaviour.
- The same limit-point conclusions hold on $B_3$ when both distinct prime divisors of $n$ are required to be odd, so the phenomena are not artifacts of the prime $2$.
Reading between the lines
- The same scheme — impose linear relations on the prime factors so that prescribed sums have prescribed largest prime factors, then count by the Green-Tao complexity-at-most-two theorem — should extend to squarefree products of $k$ primes, with interval counts of shape $X(\log X)^{-c_k}$ provided the analogous system of forms is locally soluble.
- The coexistence of a full limit-point interval with zero relative density is probably generic for functions built from largest prime factors of pairwise sums of prime divisors: the extremal values come from specially aligned parameter families, while a random three-prime product behaves as if the quotient were small.
- Theorem 3's explicit section formula is a sharp testable prediction of the paper's framework: pairs with first coordinate near $1/2$ should accumulate only at second coordinates $0$ or $9/(8m^2)$, so a computational search over $n = p^2q$ could independently confirm the fine structure before trusting the full proof.
- The $o(S(X))$ statement is purely asymptotic; at moderate $X$ the observed fraction of elements with $R(n) > \alpha$ could still be substantial, so the results do not constrain small-sample behaviour.
Original abstract
For primes $p,q,r$ that are not all equal, let $n=pqr$ and define Goldring's $w$-function by $w(n)=P(p+q)P(p+r)P(q+r)$, where $P(a)$ denotes the largest prime factor of $a$. In this paper, we resolve three problems of Chen, Shi and Wu. As $n\to\infty$ through such products, the sets of finite limit points of $P(w(n))/P(n)$ and $w(n)/n$ are $[0,1]$ and $[0,\infty)$, respectively. The same conclusions hold when $n$ is restricted either to $p^2q$ with distinct primes $p,q$ or to squarefree products of three primes. Nevertheless, on each of these three sets, the proportion of integers $n\le X$ satisfying $P(w(n))/P(n)>\alpha$ tends to zero as $X\to\infty$, for every fixed $\alpha>0$. We also determine the complete set of finite joint limit points of the two quotients for $n=p^2q$ with $p\ne q$.
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
For primes $p,q,r$ that are not all equal, let $n=pqr$ and define Goldring's $w$-function by $w(n)=P(p+q)P(p+r)P(q+r)$, where $P(a)$ denotes the largest prime factor of $a$. In this paper, we resolve three problems of Chen, Shi and Wu. As $n\to\infty$ through such products, the sets of finite limit points of $P(w(n))/P(n)$ and $w(n)/n$ are $[0,1]$ and $[0,\infty)$, respectively. The same conclusions hold when $n$ is restricted either to $p^2q$ with distinct primes $p,q$ or to squarefree products of three primes. Nevertheless, on each of these three sets, the proportion of integers $n\le X$ satisfying $P(w(n))/P(n)>\alpha$ tends to zero as $X\to\infty$, for every fixed $\alpha>0$. We also determine the complete set of finite joint limit points of the two quotients for $n=p^2q$ with $p\ne q$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | For primes $p,q,r$ that are not all equal, let $n=pqr$ and define Goldring's $w$-function by $w(n)=P(p+q)P(p+r)P(q+r)$, where $P(a)$ denotes the largest prime factor of $a$. | background 1.00 | 0.11 | 0.02 | asserted 0.92 |
| s1 | In this paper, we resolve three problems of Chen, Shi and Wu. | contribution 0.99 | 0.90 | 0.04 | asserted 1.00 |
| s2 | As $n\to\infty$ through such products, the sets of finite limit points of $P(w(n))/P(n)$ and $w(n)/n$ are $[0,1]$ and $[0,\infty)$, respectively. | result 1.00 | 0.88 | 0.65 | asserted 1.00 |
| s3 | The same conclusions hold when $n$ is restricted either to $p^2q$ with distinct primes $p,q$ or to squarefree products of three primes. | result 0.99 | 0.88 | 0.16 | asserted 1.00 |
| s4 | Nevertheless, on each of these three sets, the proportion of integers $n\le X$ satisfying $P(w(n))/P(n)>\alpha$ tends to zero as $X\to\infty$, for every fixed $\alpha>0$. | result 0.98 | 0.89 | 0.20 | asserted 1.00 |
| s5 | We also determine the complete set of finite joint limit points of the two quotients for $n=p^2q$ with $p\ne q$. | result 0.87 | 0.96 | 0.13 | asserted 1.00 |
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Discussion
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