The joint Dickman law for consecutive integers

Source and verification

This article explains family 012 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Smooth numbers and the Dickman function

Let $P^{+}(n)$ be the largest prime factor of $n$, with $P^{+}(1)=1$. An integer is $y$-smooth if all its prime factors are at most $y$.

The Dickman function is the continuous function determined by

$$\begin{array}{rlrl}\rho (u)& =1& & (0\le u\le 1),\\ u{\rho }^{\prime }(u)& =-\rho (u-1)& & (u>1).\end{array}$$

On $1\le u\le 2$, direct integration gives $\rho (u)=1-\log u$. The classical marginal law says that the proportion of integers up to $X$ with $P^{+}(n)\le X^a$ tends to $\rho (1/a)$, for fixed $0<a\le 1$.

The new issue in this family is dependence. Although $\gcd (n,n+1)=1$, this exact arithmetic relation does not prove that their largest factors are asymptotically independent.

Claimed joint law

For every fixed $a,b\in (0,1)$, the manuscript asserts

$$\begin{array}{rl}& \frac{1}{X}\#\left\{2\le n\le X:\begin{array}{c}{P}^{+}(n)\le n^a\\ P^{+}(n+1)\le n^b\end{array}\right\}\\ & ⟶\rho (1/a)\rho (1/b).\end{array}$$

The limit uses ordinary counting and every real cutoff. The parameters remain fixed, and no quantitative rate is claimed. (OpenAI, 2026)

For $a=b=1/2$, the predicted proportion is $(1-\log 2{)}^2$. This numerical expression follows from the explicit first interval of $\rho$; it is not evidence for the joint theorem.

🏷️ Why the comparison density is one half

Set

$$\begin{array}{c}{X}_n=\frac{\log P^{+}(n)}{\log n},\\ Y_n=\frac{\log P^{+}(n+1)}{\log n}.\end{array}$$

The claimed joint convergence has a product limit with continuous identical marginals. Consequently the diagonal $\{x=y\}$ has limiting probability zero. The two open triangles $x<y$ and $y<x$ have equal mass by symmetry, and together have full mass. Each therefore has mass $1/2$.

Since both coordinates use the same denominator, $X_n<Y_n$ is exactly $P^{+}(n)<P^{+}(n+1)$. Weak convergence on these continuity sets proves the ordinary density $1/2$ asserted in the source.

The absence of limiting diagonal mass is the needed argument. Symmetry of two marginals by itself would not imply this conclusion for a dependent pair.

🏷️ Encoding prime factors by finitely many phases

Fix an integer $J$. Divide primes of polynomial size into bins $(x^{k/J},x^{(k+1)/J}]$, and let ${\Omega }_k(n)$ count factors in the $k$th bin, with multiplicity. On the relevant size range, each count is bounded in terms of $J$.

For unit complex numbers ${\zeta }_k$, form

$$f_x(n)=\prod _k{\zeta }_k^{{\Omega }_k(n)}.$$

Finite Fourier inversion converts a statement about count vectors into a statement about these phase functions. The proof must allow independent phase vectors on the two neighboring integers; using the same phase would not identify their full joint law.

Center the first function by its limiting mean, writing $F_x=f_x-\mu$. The analytic target becomes

$$\frac{1}{x}\sum _{n<x}\overline{g_x(n)}{F}_x(n+1)⟶0.$$

These functions depend on $x$, and $F_x$ is no longer multiplicative. They are not automatic instances of a fixed-function correlation theorem.

🏷️ An exact change from divisors to additive shifts

Every fixed multiplier has all its prime factors below the bins once $x$ is sufficiently large. Thus multiplication by a fixed integer leaves the bin labels unchanged.

The source amplifies a hypothetical biased correlation with weighted factorizations

$$n=am,am+1=c{l}_a.$$

Cauchy–Schwarz produces two coefficients $a,b$ for the same $c,m$. If $c\mid am+1$ and $c\mid bm+1$, then $m$ is invertible modulo $c$, so $a-b=jc$ for an integer $j$. The exact identity is

$$a{l}_b-b{l}_a=\frac{a-b}{c}=j.$$

With $n^{\prime }=b{l}_a$, the two new arguments are $n^{\prime }$ and $n^{\prime }+j=a{l}_b$. Fixed-multiplier invariance transfers the original labels to these additive neighbors. This computation is the bridge from divisor amplification to an additive graph.

🏷️ Kernel approximation and the order of limits

The graph’s endpoints carry sets of auxiliary prime divisors. Randomly retaining each divisor with probability $1/2$ gives features describing the logarithmic size of the retained product. Polynomial approximation reduces these features to products

$$\prod _{p\mid n,p\in {\mathcal{P}}_B}\frac{1+p^{-v/B}}{2},$$

where $B$ is an auxiliary scale and $v\ge 0$. Weighted short-interval estimates are designed for exactly these factors.

The intermediate kernel comparison is initially valid only for tests depending separately on the endpoint sets. Actual optimizing graph tests may depend on the entire configuration. A sampling argument upgrades the estimate to expected cut norm, the supremum of normalized bilinear pairings with bounded test vectors. Moment bounds are needed because the kernels are not uniformly bounded as $B$ grows.

The original scale tends to infinity with $B$, bins, residue moduli, and feature degrees fixed. Only afterwards does $B$ increase. Short-interval cancellation then contradicts the positive energy produced by the biased subsequence. This order is what permits the source to claim an ordinary limit without exceptional scales.

🏷️ Key steps and proof outlines

Finite interpolation restores access to multiplicative estimates

The centered label $F_x=f_x-\mu$ is not multiplicative. Nevertheless, its dependence on finitely many bounded prime-bin counts allows an exact finite expansion into multiplicative functions.

Proof outline. For a single count $m\in \{0,\dots ,J\}$, choose distinct positive numbers $z_0,\dots ,z_J\le 1$. The matrix $(z_i^m{)}_{m,i}$ is invertible by the Vandermonde determinant formula. Hence any prescribed function of $m$ is an exact linear combination of $z_i^m$. Apply this separately to every bin count. Each resulting tensor-product term is nonnegative, bounded, and multiplicative because prime-factor counts add under multiplication. Centering is retained in the coefficients of this finite expansion. This construction depends only on the fixed bin partition, so its constants remain fixed as the integer scale grows.

The source’s weighted short-average lemma then separates two tasks after resolving residue restrictions by Dirichlet characters. For the principal character, approximate each short mean by its long mean; the centered linear combination of those means vanishes. For nonprincipal characters, establish small short means directly using a character-distance estimate. Summing these finitely many contributions gives the mean-square cancellation needed later, also with the divisor-feature weights $G_{B,v}$ from the kernel approximation. (OpenAI, 2026)

Auxiliary divisors must preserve a hypothetical bias

Suppose a subsequence of ordinary correlations has nonzero bias. The proof records its limiting distribution jointly with normalized position and residue classes. Write $H(t,w)$ for the resulting bounded, compactly supported profile, where $w$ represents consistent residues modulo every fixed integer. Uniform measure on these residues means that every residue modulo $q$ has probability $1/q$.

Let $D_B(w)$ be the nonnegative auxiliary divisor weight. Its mean has a positive lower bound and its $L^2$ norm is bounded uniformly in the large parameter $B$.

Proof outline. Approximate $H$ in $L^2$ by $H_q$, a function of $t$ and the residue modulo a single fixed $q$, with the same integral. For large $B$, all primes defining $D_B$ avoid $q$, so the Chinese remainder theorem gives exact independence of $D_B$ and that residue. Consequently

$$\begin{array}{rl}\int H_q{D}_B& =\left(\int H\right)\left(\int D_B\right),\\ \left|\int (H-H_q)D_B\right|& \le \Vert H-H_q{\Vert }_2\Vert D_B{\Vert }_2.\end{array}$$

First fix an accurate finite-residue approximation, then increase $B$. The second line makes the error uniformly small, while the positive mean of $D_B$ preserves the original bias. Without this lemma, multiplying by a divisor weight could destroy the very correlation being amplified.

Cauchy–Schwarz forces positive off-diagonal energy

After writing $n=am$ and $am+1=c{l}_a$, the weighted correlation has a sum over $c,m$, with $\overline{g_x(m)}$ outside an inner sum over $a$. This is legitimate because the multiplier $a$ is fixed when $x\to \infty$, so it eventually has no prime factors in the moving bins.

Proof outline. Apply weighted Cauchy–Schwarz with the nonnegative outer weights. Since $|g_x(m)|=1$, it gives $|I_{1,x}{|}^2\le V_{B,x}{I}_{2,x}$, where $I_2$ is the sum of squared inner sums and $V$ is the total outer weight. The divisor moment estimates bound $V$, and bias preservation lower-bounds $I_1$. With $T$ denoting the coefficient-scale ratio in the source’s weight, this yields

$$\underset{B\to \infty }{\mathrm{liminf}}\underset{x\to \infty }{\mathrm{liminf}}\frac{I_{2,x}}{BT}\ge c_5>0.$$

The diagonal terms $a=b$ have normalized mass tending to zero, so this positivity must survive in the off-diagonal terms. For those terms the common factorization gives $c\mid (a-b)$ and, with $j=(a-b)/c$,

$$b{l}_a=n^{\prime },a{l}_b=n^{\prime }+j.$$

Thus a positive divisor energy becomes an additive-shift energy of the centered labels. This is why the exact identity derived earlier is useful: it transfers an energy with a proven positive lower bound into the setting of short-average cancellation.

Sampling makes globally chosen signs into local tests

The kernel estimate initially controls test functions depending on a single endpoint. In a cut norm, however, the maximizing signs may depend on the entire random matrix. One cannot condition on those signs and still assume the endpoint types are independent.

Proof outline. For a fixed realized $M\times M$ error matrix $E$, choose optimizing column signs ${\delta }_k$ and put $r_i={\sum }_k{E}_{ik}{\delta }_k$. Sample $m$ columns without replacement and estimate

$${\hat{r}}_i=\frac{M}{m}\sum _{j\in J^{\prime }}{E}_{ij}{\delta }_j.$$

The estimate is unbiased and has variance at most $(M/m){\sum }_k{E}_{ik}^2$. The elementary inequality $|r|-r\mathrm{sgn}(\hat{r})\le 2|r-\hat{r}|$, followed by Cauchy–Schwarz, bounds the loss from replacing optimal row signs by $\mathrm{sgn}({\hat{r}}_i)$. Hence some small sample incurs little loss. Delete all edges meeting that sample; degree bounds make this inexpensive. Now fix the sample and its signs, and condition only on its endpoint types. Each surviving row sign depends on its own site and the fixed sampled data, so the one-site kernel estimate applies. Uniform conditional estimates and the source’s concentration argument handle the possible sample choices. The second-moment and degree truncations are essential because the untruncated kernels need not be bounded uniformly in $B$.

Positive energy contradicts the weighted short-average estimate

The endpoint kernels encode randomly retaining each auxiliary prime with probability $1/2$. Approximating their features by finitely many polynomials produces the multiplicative factors

$$G_{B,v}(n)=\prod _{p\mid n,p\in {\mathcal{P}}_B}\frac{1+p^{-v/B}}{2}.$$

Partitioning into fixed residue classes and position blocks turns the resulting energy terms into products of short averages of $F_x{G}_{B,v}$. The first lemma makes their averaged squared magnitudes tend to zero. Approximation, sampling, and discarded-diagonal errors are chosen small compared with $c_5$, contradicting the positive lower bound.

flowchart TD
  A[Hypothetical ordinary correlation bias] --> B[Divisor weighting preserves bias]
  B --> C[Cauchy-Schwarz gives positive off-diagonal energy]
  C --> D[Exact factorization identity gives additive shifts]
  E[Finite multiplicative interpolation] --> F[Weighted short-average cancellation]
  G[Sampling and kernel approximation] --> H[Express energy using short averages]
  D --> H
  F --> I[The same normalized energy tends to zero]
  H --> I
  C --> J[Contradiction]
  I --> J

Throughout this diagram, the integer scale tends to infinity with $B$ and all finite approximation data fixed. The auxiliary limits follow afterwards. Finally finite Fourier inversion recovers the prime-bin count distribution. To replace fixed thresholds $X^a,X^b$ by $n^a,n^b$, restrict to $\epsilon X\le n\le X$, sandwich the logarithmic thresholds there, and use continuity of the Dickman distribution before sending $\epsilon \downarrow 0$.

🏷️ Connections

Ordinary multiplicative correlations provides the related distinction between fixed-shift ordinary and logarithmic averages. Prime predecessor partitions explains the full ranked-factor law behind the Dickman marginal.

🏷️ Manuscripts and verification scope

• The joint Dickman law for consecutive integers — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The page selects the ordinary joint law and the two ordering densities of one half. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. The joint Dickman law for consecutive integers.