Elliott’s conjecture

For every integer $k\ge 1$ , every collection of distinct shifts $h_1,\ldots,h_k\in\mathbb{Z}$ , and every $1$ -bounded multiplicative functions $f_1,\ldots,f_k:\mathbb{N}\to\mathbb{C}$ , if at least one function, say $f_j$ , is non-pretentious in the sense that for every Dirichlet character $\chi$ and every $t\in\mathbb{R}$ , $\sum_{p}\frac{1-\operatorname{Re}\!\left(f_j(p)\overline{\chi(p)}p^{-it}\right)}{p}=\infty$ , then $\frac{1}{x}\sum_{n\le x}\prod_{i=1}^{k}f_i(n+h_i)\longrightarrow 0$ as $x\to\infty$ . The original formulation is false in general; stronger corrected versions remain open.

References

Primary source

arXiv

Additional references

Truncated pretentious distances of multiplicative arithmetic functions — arXiv — Thomas Renard

analytic-number-theory multiplicative-functions

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed progress

A new paper extends Elliott’s conjecture to another class of multiplicative functions, but the corrected general conjecture remains open.

Elliott’s conjecture concerns correlations of multiplicative functions. Its original complex-valued formulation is false, while corrected strongly non-pretentious versions remain open.

Known results

Matomäki, Radziwiłł, and Tao (2015-era work): constructed counterexamples to the original formulation and proposed corrected versions.
Tao and Teräväinen (2018): established structural correlation results at almost all scales and several special cases.
Tao and Teräväinen (2023): proved further correlation results for non-pretentious functions, including higher-order cases and Liouville-like applications.

September 29, 2026 partial advance

Thomas Renard’s paper Truncated pretentious distances of multiplicative arithmetic functions claims convergence of relevant autocorrelation averages for a newly identified class of non-pretentious multiplicative functions. This is a claimed partial advance, not a resolution of Elliott’s conjecture.

Current status (as of September 2026): The original formulation is known to be false, several corrected cases are proved, and Renard’s new extension is unverified; the general corrected conjecture remains open.

Sources

No solutions have been posted yet.