Ordinary two-point correlations and the corrected Elliott conjecture

1 · The essentials

The problem compares the values of an arithmetic rule at two positions that move together. Those positions follow two fixed expressions of the form “a multiple of the integer, plus a constant,” which must not be proportional. For the rule called the Liouville function, the manuscript shows that the products of the paired values largely cancel when added together. It gives a bound at every summation cutoff, not just at selected cutoffs. A broader result concerns two different arithmetic rules whose values may be complex numbers of size at most one. It requires at least one of them not to resemble certain precisely specified reference rules too closely. The main result has a Lean formalization, but the supplied information does not specify its exact scope.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine two readers moving through a long list according to two fixed movement instructions and combining the values they encounter. The result says that the total of those combinations stays small compared with the number of readings, under the stated conditions; it does not say that each individual encounter cancels out.

Why it matters

The manuscript addresses ordinary averages, in which every summand receives the same weight. For the Liouville function, it also announces a quantitative estimate at every cutoff. The general result identifies a condition under which vanishing correlation extends to a broader class of functions.

2 · Going further

Consider two fixed affine forms $L_{1} (n) = a_{1} n + b_{1}$ and $L_{2} (n) = a_{2} n + b_{2}$ that are not proportional, on integers where the functions are defined. For the Liouville function $λ$ , the announced result is a bound of the form $\sum_{n \leq X} λ (L_{1} (n)) λ (L_{2} (n)) = O (X / (lo g X)^{c})$ , with an absolute exponent $c > 0$ . The abstract specifies that this power-of-logarithm saving holds at every summation cutoff.

The extension concerns complex multiplicative functions of modulus at most one. The manuscript shows the binary corrected Elliott conjecture for ordinary averages under uniform nonpretentiousness of at least one original factor. The repository summary describes this hypothesis as uniform non-resemblance to each fixed Dirichlet character multiplied by $n^{i t}$ , for $∣ t ∣ \leq X$ . The qualitative conclusion also holds in fixed residue classes and for fixed nonproportional affine forms.

The supplied context is the two-point Chowla conjecture and the binary corrected Elliott conjecture. The excerpt does not describe the idea of the argument, so its steps cannot be reconstructed from this source alone. Only one main manuscript is presented, with no information about companion manuscripts. A Lean formalization of the main result exists, but its detailed scope is not specified here.

Useful notions

Sums and averages
Affine forms and proportionality
Multiplicative functions
Complex numbers and modulus
Congruences and Dirichlet characters

What was known before / what is new

Before

The source places the work in the context of the two-point Chowla conjecture and the binary corrected Elliott conjecture. It supplies neither a history of these problems nor an account of earlier partial results.

New

For Liouville, the manuscript announces a bound $O (X / (lo g X)^{c})$ with absolute $c > 0$ , at every cutoff and for every fixed pair of nonproportional affine forms. It also shows a qualitative conclusion for two complex multiplicative functions bounded in modulus by one, under the stated uniform nonpretentiousness condition. This extension includes fixed residue classes.

Limits and assumptions

The affine forms and residue classes are fixed; proportional forms are excluded. The general result assumes multiplicative functions of modulus at most one and uniform nonpretentiousness of at least one original factor against each fixed Dirichlet character multiplied by $n^{i t}$ for $∣ t ∣ \leq X$ . The announced quantitative bound concerns Liouville; the general extension is described as qualitative. The main result has a Lean formalization, but the source does not establish that this verification covers every extension or detail. The excerpt does not specify which questions remain open.

Mini glossary

Two-point correlation: A sum or average of products of two function values taken along two specified expressions.
Ordinary average: An average in which all the terms under consideration receive equal weight.
Affine form: An expression of the form $an + b$ , with fixed coefficients.
Uniform nonpretentiousness: A condition of non-resemblance to the specified reference functions, imposed uniformly over the parameter range under consideration.
Power-of-logarithm saving: A reduction of a bound of size $X$ by a factor $(lo g X)^{c}$ , with $c > 0$ .

Keywords : Chowla · Corrected Elliott · Ordinary correlations

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves the ordinary two-point Chowla conjecture, with a bound $O (X / (lo g X)^{c})$ for Liouville correlation sums along fixed nonproportional affine forms, where c > 0 is absolute. More generally, proves the binary corrected Elliott conjecture for complex multiplicative functions bounded by one when one factor is uniformly nonpretentious against each fixed Dirichlet character times $n^{i t}$ for $∣ t ∣ \leq X$ .

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/007.md

Manuscripts in this family

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Ordinary two-point correlations of multiplicative functions

24/09/2026

To be verified

We prove the ordinary two-point Chowla conjecture. For every fixed pair of nonproportional affine forms, the Liouville correlation has a power-of-logarithm saving at every cutoff, with an absolute exponent. We also prove the binary corrected Elliott conjecture for ordinary averages of complex multiplicative functions of modulus at most one, under uniform nonpretentiousness of at least one original factor. This qualitative conclusion holds in fixed residue classes and for fixed nonproportional affine forms.

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