Proceedings of the London Mathematical Society· 2026Q1
On Elliott's conjecture and applications
- 2citations
- Q1SCImago
- 2026year
Short summary
A new proof shows that for nonpretentious multiplicative functions, the two-point correlation vanishes for distinct integer shifts along a set of full upper logarithmic density, a significant improvement over previous work requiring stronger hypotheses.
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Key points
- The two-point correlation of nonpretentious multiplicative functions vanishes for distinct integer shifts on a set of full upper logarithmic density.
- This result is achieved under the minimal assumption of nonpretentiousness, relaxing previous stronger hypotheses.
- The proof is extended to odd-point correlations for real-valued nonpretentious functions.
- Applications include solving R.H. Hudson's 1974 conjecture and a problem by de la Rue on the Elliott conjecture.
AI-generated from the title and abstract; the full text is not read.
Abstract
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.
The authors' abstract, as published at the source. Proceedings of the London Mathematical Society, 2026 · DOI ↗
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Field: Algebra and Number Theory
Algebra and Number TheoryMathematics