## Duke Mathematical Journal

Published by Duke University Press since its inception in 1935, the *Duke Mathematical Journal* is one of the world's leading mathematical journals. *DMJ* emphasizes the most active and influential areas of current mathematics. Advance publication of articles online is available.

Learn about DMJ's founding and visit DMJ By the Numbers for key facts about this flagship journal.

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**ISSN:**0012-7094 (print), 1547-7398 (electronic)**Publisher:**Duke University Press**Discipline(s):**Mathematics**Full text available in Euclid:**1935--**Access:**By subscription only**Euclid URL:**https://projecteuclid.org/dmj

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*MR Citation Database* MCQ (2018): 2.79

*JCR* (2018) Impact Factor: 2.199

*JCR* (2018) Five-year Impact Factor: 2.766

*JCR* (2018) Ranking: 15/313 (Mathematics)

Article Influence (2018): 4.346

Eigenfactor: Duke Mathematical Journal

SJR/SCImago Journal Rank (2018): 5.73

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### Featured article

* *The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures

##### Abstract

Let ${g}_{0},\dots ,{g}_{k}:\mathbb{N}\to \mathbb{D}$ be $1$-bounded multiplicative functions, and let ${h}_{0},\dots ,{h}_{k}\in \mathbb{Z}$ be shifts. We consider correlation sequences $f:\mathbb{N}\to \mathbb{Z}$ of the form $$f(a):=\underset{m\to \infty}{}\frac{1}{log{\omega}_{m}}{\sum}_{{x}_{m}/{\omega}_{m}\le n\le {x}_{m}}\frac{{g}_{0}(n+a{h}_{0})\cdots {g}_{k}(n+a{h}_{k})}{n},$$ where $1\le {\omega}_{m}\le {x}_{m}$ are numbers going to infinity as $m\to \infty $ and $$ is a generalized limit functional extending the usual limit functional. We show a structural theorem for these sequences, namely, that these sequences $f$ are the uniform limit of periodic sequences ${f}_{i}$. Furthermore, if the multiplicative function ${g}_{0}\cdots {g}_{k}$ “weakly pretends” to be a Dirichlet character $\chi $, the periodic functions ${f}_{i}$ can be chosen to be $\chi $-isotypic in the sense that ${f}_{i}\left(ab\right)={f}_{i}\left(a\right)\chi \left(b\right)$ whenever $b$ is coprime to the periods of ${f}_{i}$ and $\chi $, while if ${g}_{0}\cdots {g}_{k}$ does not weakly pretend to be any Dirichlet character, then $f$ must vanish identically. As a consequence, we obtain several new cases of the logarithmically averaged Elliott conjecture, including the logarithmically averaged Chowla conjecture for odd order correlations. We give a number of applications of these special cases, including the conjectured logarithmic density of all sign patterns of the Liouville function of length up to three and of the Möbius function of length up to four.

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