On Elliott's conjecture and applications
- 1School of Mathematics University of Bristol Bristol UK
- 2Department of Mathematical Sciences Durham University Upper Mountjoy Campus Durham UK
- 3Department of Pure Mathematics and Mathematical Statistics University of Cambridge Cambridge UK
Abstract
Let be a multiplicative function. Under the merely necessary assumption that is nonpretentious (in the sense of Granville and Soundararajan), we show that for any pair of distinct integer shifts , the two‐point correlation tends to 0 along a set of of full upper logarithmic density. We also show that the same result holds for the ‐point correlations if is odd and is a real‐valued nonpretentious function. Previously, the vanishing of correlations was known only under stronger nonpretentiousness hypotheses on by the works of Tao, and Tao and the third author. We derive several applications, including: A classification of ‐valued completely multiplicative functions that omit a length four sign pattern, solving a 1974 conjecture of R.H. Hudson. A proof that a class of “Liouville‐like” functions satisfies the unweighted Elliott conjecture of all orders, solving a problem of de la Rue. Constructing examples of multiplicative with a given (unique) Furstenberg system, answering a question of Lemańczyk. A density version of the Erdős discrepancy theorem of Tao.
Related
- On the finite field spherical restriction conjecture in four dimensions: the sharp endpoint and applications
- A uniform lower bound for the Zhang-Kawazumi invariant and applications to the Bogomolov conjecture
- An Equivariant Main Conjecture for Branched ℤp-Towers of Finite Graphs and Applications
- An explicit form of the standard conjecture of Hodge type for Hermitian varieties, with p-adic applications
- Solving the Ho--Nguyen Conjecture for Gaussian Mixtures: Applications to Mixtures of Experts
BibTeX
@article{klurman2026elliotts,
title = {{On Elliott's conjecture and applications}},
author = {Oleksiy Klurman and Alexander P. Mangerel and Joni Teräväinen},
journal = {Proceedings of the London Mathematical Society},
year = {2026},
month = sep,
doi = {10.1112/plms.70205},
url = {https://doi.org/10.1112/plms.70205},
note = {CC BY},
}
RIS
TY - JOUR
TI - On Elliott's conjecture and applications
AU - Oleksiy Klurman
AU - Alexander P. Mangerel
AU - Joni Teräväinen
PY - 2026
DA - 2026/09/01
JO - Proceedings of the London Mathematical Society
DO - 10.1112/plms.70205
UR - https://doi.org/10.1112/plms.70205
AB - Let be a multiplicative function. Under the merely necessary assumption that is nonpretentious (in the sense of Granville and Soundararajan), we show that for any pair of distinct integer shifts , the two‐point correlation tends to 0 along a set of of full upper logarithmic density. We also show that the same result holds for the ‐point correlations if is odd and is a real‐valued nonpretentious function. Previously, the vanishing of correlations was known only under stronger nonpretentiousness hypotheses on by the works of Tao, and Tao and the third author. We derive several applications, including: A classification of ‐valued completely multiplicative functions that omit a length four sign pattern, solving a 1974 conjecture of R.H. Hudson. A proof that a class of “Liouville‐like” functions satisfies the unweighted Elliott conjecture of all orders, solving a problem of de la Rue. Constructing examples of multiplicative with a given (unique) Furstenberg system, answering a question of Lemańczyk. A density version of the Erdős discrepancy theorem of Tao.
SN - 0024-6115
SN - 1460-244X
ER -