## 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.

Propagation in Hamiltonian dynamics and relative symplectic homologyVolume 119, Number 1 (2003)
Arithmetic of double torus quotients and the distribution of periodic torus orbitsVolume 168, Number 12 (2019)
The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjecturesVolume 168, Number 11 (2019)
Conserved energies for the cubic nonlinear Schrödinger equation in one dimensionVolume 167, Number 17 (2018)
Weight elimination in Serre-type conjecturesVolume 168, Number 13 (2019)
• 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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### Featured article

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

Volume 168, Number 11 (2019)
##### 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 $\begin{equation*}f(a):=\mathop{\widetilde{\lim }}_{m\to \infty }\frac{1}{\log \omega_{m}}\sum _{x_{m}/\omega _{m}\leq n\leq x_{m}}\frac{g_{0}(n+ah_{0})\cdots g_{k}(n+ah_{k})}{n},\end{equation*}$ where $1\leq \omega _{m}\leq x_{m}$ are numbers going to infinity as $m\to \infty$ and $\mathop{\widetilde{\lim }}$ 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 $f_{i}(ab)=f_{i}(a)\chi (b)$ 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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