Arbitrarily long runs of equal signs of the Liouville function

The Liouville function $\lambda(n)$ is $+1$ or $-1$ according to whether $n$ has an even or odd number of prime factors, and it is expected to behave like a sequence of fair coin flips. Random coin flips contain arbitrarily long runs of heads. Can one prove that $\lambda$ takes the same value on arbitrarily many consecutive integers? Runs of length four are known; arbitrary lengths are not.

Problem

Let $\Omega(n)$ be the number of prime factors of $n$ counted with multiplicity and $\lambda(n) = (-1)^{\Omega(n)}$. For every $k$, are there integers $n$ with

$$\lambda(n) = \lambda(n+1) = \cdots = \lambda(n+k-1) = +1,$$

and likewise with all values equal to $-1$?

By the source's remark, an approach through the entropy decrement argument roughly reduces this to finding arbitrarily long progressions $n, n+p, \dots, n+kp$ on which $\lambda$ is constant.

Background

The question appears in the problem list of the AIM workshop on Sarnak's conjecture. Chowla's conjecture predicts that every sign pattern of every length occurs in $\lambda$ with the frequency of a random sequence, and Sarnak's conjecture is a weaker form concerning correlations with low-complexity sequences. Patterns of $\lambda$ at consecutive integers are hard because sieve methods cannot see the parity of the number of prime factors; progress since 2015 has come from the Matomäki–Radziwiłł theorem on multiplicative functions in short intervals and from Tao's entropy decrement argument.

What is known

• Hildebrand (1986) proved that all eight sign patterns of length three occur infinitely often in $(\lambda(n), \lambda(n+1), \lambda(n+2))$; Matomäki, Radziwiłł and Tao (2016) showed that each occurs with positive lower density.
• Tao and Teräväinen (2019) proved that all 16 sign patterns of length four occur, so runs of four equal values exist.
• For length five, Tao and Teräväinen (2019) showed that at least 24 of the 32 sign patterns occur with positive upper density. The twelve patterns they exhibit explicitly (six and their reversals) are not constant; whether the two constant patterns of length five occur is not known, and for longer lengths much less is known.
• Matomäki, Radziwiłł, Tao, Teräväinen and Ziegler (2023) showed that the number of sign patterns of the Liouville function is superpolynomial, without identifying which patterns occur.

What would count as progress

• Runs of any fixed length beyond four: for example, a proof that five consecutive equal values of $\lambda$ occur infinitely often.
• A proof for arbitrary $k$, possibly along the reduction to monochromatic progressions $n, n + p, \dots, n + kp$ mentioned in the source.
• Conditional results: deriving long runs from a clearly stated hypothesis weaker than Chowla's conjecture.

References

1. K. Matomäki, M. Radziwiłł and T. Tao, Sign patterns of the Liouville and Möbius functions, Forum Math. Sigma 4 (2016), e14; arXiv:1509.01545.
2. T. Tao and J. Teräväinen, The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures, Duke Math. J. 168 (2019), 1977–2027.
3. T. Tao and J. Teräväinen, Value patterns of multiplicative functions and related sequences, Forum Math. Sigma 7 (2019), e33; arXiv:1904.05096.
4. K. Matomäki, M. Radziwiłł, T. Tao, J. Teräväinen and T. Ziegler, Higher uniformity of bounded multiplicative functions in short intervals on average, Ann. of Math. 197 (2023), 739–857.

Source

Imported from AIM Problem Lists sarnakconjecture 2.2, curated by American Institute of Mathematics. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.