Ordinary two-point correlations and the corrected Elliott conjecture

The explanation

Take the number 12. It is 2 × 2 × 3, so it has three prime factors when repeats count. Three is odd. Now take 10, which is 2 × 5. It has two prime factors, an even count. Give every whole number a sign by this rule: +1 for an even count, −1 for an odd one. This sign is called the Liouville function, written λ(n). From 1 to 12 the signs run +, −, −, +, −, +, −, −, +, +, −, −.

Now compare each number with the next one. The numbers 9 and 10 share a sign. The numbers 10 and 11 do not. Among the first ten neighbouring pairs, only three agree. That looks like a pattern, but it fades. Up to 10 million, neighbours agree 4,998,976 times and disagree 5,001,024 times. The difference is about 0.02 percent of all pairs.

In 1965, Sarvadaman Chowla conjectured that this is no accident. Fix any shift h, such as 1 or 7. His conjecture says the signs of n and n + h become uncorrelated. In the long run they agree as often as they disagree. The version for one pair of positions is called the two-point case. Peter Elliott extended the question in 1992 to a wider class of functions. These multiplicative functions obey f(mn) = f(m)f(n) whenever m and n share no prime factor.

The difficulty is the kind of average. A plain average up to some cutoff X gives every number weight one. A logarithmic average gives the number n weight 1/n, so small numbers count more. That blend mixes information from many scales of size at once. A plain average must control the last stretch below X directly. A plain bound implies the logarithmic one, but not the other way round.

Progress stopped at exactly that gap. In 2016, Kaisa Matomäki and Maksym Radziwiłł showed that each such correlation stays below its largest possible size by some margin. Also in 2016, Terence Tao proved the two-point case for logarithmic averages. Harald Helfgott and Radziwiłł in 2021, and then Cédric Pilatte, proved strong rates for that weighted sum. For plain averages, cancellation was known only at almost all scales, not at every cutoff.

This paper proves the plain version at every cutoffSource 1. One fact makes the method natural. Multiply both numbers in a pair by the same prime, and both signs flip. Their product stays the same. So the pair n and n + h tells you about the pair 3n and 3n + 3h. These comparisons link whole numbers into a large network. The proof controls the round trips in it. That idea comes from Helfgott and Radziwiłł's network method and Pilatte's refinements. The paper adapts it so every number weighs the same.

For the Liouville signs, the paper also gives a rateSource 2. The gap between agreements and disagreements up to X is at most a constant times X/(log X)^c. Here c is one fixed positive number for every pair of forms. The same holds along straight-line patterns such as 3n + 2 and nSource 2. The rate is slow, because log X grows slowly: at 10 million it is only about 16. The constants are not given explicitly, so the theorem does not predict the 0.02 percent at 10 million. It says the share must shrink toward zero.

For the wider class of multiplicative functions, the paper proves plain-average cancellation with no rate. It needs one condition on at least one of the two functions. At every large scale, it must stay far from every simple wave-like pattern. The pattern may even change from scale to scale. The paper calls this uniformly nonpretentious. It also claims nothing when the shifts or coefficients grow with XSource 3.

How sure is this? A computer proof checker called Lean has checked formal versions of both main statements. They are the plain two-point cancellation and the Liouville rate. A tool called Comparator matches those checks against fixed reference statements. The paper itself has no published specialist review yet.

What it doesn't show. This does not settle the conjecture for three or more shifts at once.