Two-point Chowla: do neighboring integers forget prime-factor parity?

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Family 007 · Number theory
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

Count the prime factors of a positive integer, including repetitions, and record whether that count is even or odd. For example, $12=2^2\cdot3$ has three prime factors and $10=2\cdot5$ has two. Does the parity for $n$ give any persistent information about the parity for $n+1$?

The two-point Chowla conjecture predicts that the two parities become uncorrelated when averaged over longer and longer initial intervals. The same question can be asked about $n$ and $n+h$ for any fixed positive shift $h$, or about two nonproportional linear expressions in $n$.

Family 007’s manuscript claims ordinary two-point cancellation at every sufficiently large scale, with a power-of-logarithm saving for the Liouville function. It also claims a broader qualitative theorem for bounded multiplicative functions that do not imitate a periodic character times a smooth multiplicative oscillation.

The word “ordinary” carries much of the significance. Each integer receives equal weight. Earlier theorems established logarithmically weighted cancellation, averages over shifts, or conclusions outside exceptional sets of scales. Those are major partial results, but they do not by themselves give the asserted every-cutoff conclusion.

2. Background and essential definitions

Let $\Omega(n)$ count prime factors with multiplicity, and define the Liouville function by

$$\lambda(n)=(-1)^{\Omega(n)}.$$

Thus $\lambda(12)=-1$, $\lambda(10)=1$, and $\lambda(1)=1$. It is completely multiplicative:

$$\lambda(mn)=\lambda(m)\lambda(n)$$

for all positive integers $m,n$, even when they share factors.

The correlation

$$\frac1X\sum_{1\le n\le X}\lambda(n)\lambda(n+h)$$

is the difference between the proportions of equal and unequal parities, up to the harmless use of $X$ rather than $\lfloor X\rfloor$. Tending to zero means these two possibilities balance asymptotically. It does not make the deterministic sequence literally random.

Compare ordinary and logarithmic averages of a bounded sequence $u(n)$:

$$\frac1X\sum_{n\le X}u(n), \qquad \frac1{\log X}\sum_{n\le X}\frac{u(n)}n.$$

The logarithmic average gives comparable total weight to intervals whose endpoints differ by the same multiplicative factor. An ordinary average is much more sensitive to what happens near its final scale. Cancellation in the ordinary sense implies logarithmic cancellation by summation, but the reverse implication is unavailable in general.

A function $f$ is multiplicative if $f(mn)=f(m)f(n)$ whenever $m,n$ are coprime. This weaker condition permits independent choices at prime powers. The manuscript allows complex values with $|f(n)|\le1$, including zero.

Some such functions have obvious structure. A Dirichlet character $\chi$ is multiplicative and periodic, with root-of-unity values away from its modulus. The function $n^{it}=e^{it\log n}$, for real $t$, is also multiplicative. Their products are natural obstructions to blanket cancellation theorems.

To measure imitation, define the nonnegative distance

$$\mathbb D(f,\chi n^{it};X)^2 =\sum_{p\le X} \frac{1-\operatorname{Re}\bigl(f(p)\overline{\chi(p)}p^{-it}\bigr)}p,$$

where the sum is over primes. Uniform nonpretentiousness means that for every fixed character $\chi$,

$$\inf_{|t|\le N}\mathbb D(f,\chi n^{it};N) \longrightarrow\infty.$$

The possible twist $t$ varies with the scale. Excluding only each fixed $t$ separately would be a weaker hypothesis.

3. Development history and previous partial results

Chowla proposed a general vanishing principle for products of Liouville values at distinct fixed shifts. The binary case is already difficult because multiplicativity controls products of arguments, while the question compares additive shifts.

Elliott extended the problem to correlations of bounded multiplicative functions. Work of Matomäki, Radziwiłł, and Tao identified a necessary correction for complex functions: a function can resemble different twists at successive scales without resembling one fixed twist globally. Their averaged form of Chowla’s conjecture explains the corrected condition and proves cancellation averaged over ranges of shifts.

Matomäki and Radziwiłł’s short-interval theorem showed that most short averages of a bounded real multiplicative function approximate its long average. They also obtained a strict improvement over the trivial fixed-shift correlation bound, without showing the normalized correlation tends to zero at every cutoff.

Tao’s logarithmically averaged theorem established the binary corrected Elliott conclusion with logarithmic weights, including nonproportional affine forms. Helfgott and Radziwiłł then developed divisibility graphs and quantitative expansion estimates. Pilatte’s work improved the saving for the logarithmically weighted two-point Liouville correlation to a power of a logarithm.

For ordinary averages, Tao and Teräväinen obtained binary cancellation outside a set of scales of logarithmic density zero. Later work of Klurman, Mangerel, and Teräväinen allowed a weaker nonpretentiousness assumption along a suitable set of scales. M1 claims to remove exceptional cutoffs under the uniform condition stated above, and to give a quantitative Liouville bound for fixed affine forms.

The manuscript explicitly builds on these graph and short-interval methods. Its claimed advance lies in the finite-scale estimates and transfers needed for ordinary averages, rather than in introducing multiplicative correlations from scratch.

4. What the new manuscripts claim

A quantitative Liouville theorem

M1, Theorem 1.1, asserts the existence of an absolute $c>0$ with the following property. Fix integers

$$a_1,a_2\ge1,\qquad b_1,b_2\ge0, \qquad a_1b_2-a_2b_1\ne0.$$

Then there is a constant $C$, depending on these coefficients, such that for every real $X\ge3$,

$$\left|\sum_{1\le n\le X} \lambda(a_1n+b_1)\lambda(a_2n+b_2)\right| \le C\frac{X}{(\log X)^c}.$$

The exponent is common to all fixed forms. The coefficient-dependent constants need not be effective, and the theorem claims no uniformity when the coefficients grow with $X$. No coprimality condition on the coefficients is imposed.

The determinant condition excludes proportional forms. This is essential: if the two forms are $n$ and $2n$, then complete multiplicativity gives $\lambda(n)\lambda(2n)=-1$ identically, so cancellation is impossible.

A qualitative theorem for general functions

Theorem 1.2 states that if $f_1,f_2$ are multiplicative, $|f_j(n)|\le1$, and at least one original factor is uniformly nonpretentious, then for distinct fixed nonnegative shifts $h_1,h_2$,

$$\frac1N\sum_{n=1}^N f_1(n+h_1)f_2(n+h_2)\longrightarrow0.$$

There is no conjugate in this product. A conjugated version follows by applying the theorem to a conjugated function. Corollary 1.3 gives the same qualitative conclusion for the fixed nonproportional affine forms above.

The manuscript also allows any fixed residue-class restriction, including classes not coprime to the modulus. The modulus and class remain fixed as the cutoff grows. The condition is imposed on an original multiplicative function, not merely on an expression obtained after changing variables.

Finally, the Möbius function $\mu$ equals $\lambda$ on squarefree integers and is zero otherwise. Since $\mu(p)=\lambda(p)=-1$ at primes, the distance condition is the same for both. M1 therefore deduces qualitative two-point Möbius and mixed Möbius–Liouville cancellation. It does not claim the Liouville logarithmic rate for these products containing a Möbius factor.

5. Why the advance matters

Provisional importance: 5/5. Ordinary two-point Chowla is a central test of the principle that multiplicative arithmetic can behave randomly under additive shifts. An every-cutoff theorem would remove a persistent gap between several strong averaged results and the original correlation question.

The affine and general-function formulations broaden the scope beyond neighboring integers. They describe when multiplicative structure can survive in a shifted correlation and when absence of a specific kind of structure forces cancellation.

The quantitative distinction also matters: a fixed logarithmic saving provides a definite asymptotic bound for Liouville, while the general theorem guarantees convergence with no common speed. The rating concerns the significance of these stated conclusions if correct, rather than the apparent randomness of numerical experiments.

6. The main idea

Build a graph from multiplication and divisibility

Multiplicativity relates the values at $n,n+h$ to values at $dn,d(n+h)$. The latter arguments differ by $hd$. The manuscript organizes such comparisons as a weighted graph on integers, with steps determined by products of selected primes.

For a prime $p$, the indicator $\mathbf1_{p\mid n}$ has average approximately $1/p$. Subtracting that mean produces the centered quantity

$$\mathbf1_{p\mid n}-\frac1p.$$

Centering makes cancellation available when a prime is averaged independently. Short-interval Fourier estimates control the error introduced when the original divisibility weights are replaced by these signed expressions.

A graph is encoded by a matrix. Large powers of that matrix count closed walks, and bounds for their traces can control the size of the associated operator. In arithmetic graphs, repeated prime labels create congruence constraints. Walks with many independent constraints are rare. When fewer constraints remain, the manuscript organizes the repetition patterns with forests, keeping the number of possible patterns under control.

Obtain a saving that survives at the final scale

For Liouville, each graph step takes one prime from each of $J$ disjoint supplies. The total useful weight grows roughly like $W^J$, whereas the graph bound has size roughly $(C\sqrt W)^J$, with a large fixed $W$. Choosing $W$ sufficiently large gives exponential decay in $J$.

The construction takes $J$ proportional to $\log\log X$. Exponential decay in $J$ then becomes a negative power of $\log X$, explaining the form of the headline saving. Extra disjoint prime factors control how divisor weight is distributed among short multiplicative intervals.

A finite interval of integers is too short to average over the full joint period of all selected primes. M1 instead compares the specific residue tests used in the argument with an independent-residue model. It invokes bounded-independence results from Boolean circuit theory after correcting the encoded distribution to exact bounded independence. This step is what allows the comparison to remain quantitative at the actual cutoff.

Why the general theorem uses a different limit

Uniform nonpretentiousness may diverge arbitrarily slowly. The general-function argument therefore fixes an auxiliary graph scale $B$, lets the ordinary averaging interval grow, and only afterward lets $B$ increase.

A hypothetical persistent correlation would force a graph average of size comparable to $B^{-1}$. The analytic replacement error is smaller, while the graph estimate gains an additional fixed negative power of $B$. Increasing $B$ contradicts the original bias.

The weights here differ from the Liouville construction. Certain signed divisibility factors are not individually mean zero; the needed cancellation occurs only after including normalization factors from both endpoints of a walk. M1 keeps those prime residues unconditioned until the signed average is taken. Taking absolute values too early would lose the effect the strategy relies on.

These are two related graph arguments with different quantitative information, not one rate estimate automatically valid for every multiplicative function.

7. Scope, limitations, and open questions

The manuscript addresses two-point correlations. It does not settle Chowla’s full conjecture for every number of distinct shifts. Pairwise cancellation does not imply cancellation of every longer product.

All shifts, affine coefficients, and residue moduli in the conclusions are fixed. The every-cutoff assertion should not be interpreted as a uniform theorem for parameters increasing with the cutoff. The positive logarithmic exponent is absolute, but the implied constants may be ineffective.

The general corrected Elliott statement needs uniform nonpretentiousness over a growing twist range. A hypothesis excluding only fixed twists would not be the theorem stated here. Neither the Möbius specialization nor arbitrary bounded multiplicative functions inherit a quantitative rate from the Liouville result.

Natural further questions concern higher correlations, stronger savings, and simultaneous growth of the shifts or moduli. They remain beyond this family’s stated conclusions.

8. Sources and further reading

M1’s Theorems 1.1–1.2 and Corollary 1.3 give the main claims. Its introductory overview distinguishes the two graph arguments; Proposition 18.4 treats progressions, and Section 18.3 explains the Liouville and Möbius specializations.

The primary papers in Section 3 trace the short-interval, logarithmic, and almost-all-scale results. For optional notation, see the Liouville function and multiplicative functions. The essential averaging and nonpretentiousness conventions are stated explicitly above.

Source manuscripts

• M1. OpenAI, Ordinary two-point correlations of multiplicative functions (September 24, 2026).

Sources consulted on 10 October 2026 at repository revision fd4aeeb. Download the manuscript citations.