## Publication Date:

23 July 2019

## Journal:

Duke Mathematical Journal

## Last Updated:

2021-09-24T23:54:14.007+01:00

## Issue:

11

## Volume:

168

## DOI:

10.1215/00127094-2019-0002

## page:

1977-2027

## abstract:

Let $g_0,\dots,g_k: {\bf N} \to {\bf D}$ be $1$-bounded multiplicative

functions, and let $h_0,\dots,h_k \in {\bf Z}$ be shifts. We consider

correlation sequences $f: {\bf N} \to {\bf Z}$ of the form $$ f(a):=

\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) \dots g_k(n+ah_k)}{n} $$ where $1 \leq \omega_m

\leq x_m$ are numbers going to infinity as $m \to \infty$, and

$\widetilde{\lim}$ is a generalised 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 $g_0 \dots g_k$ "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 $g_0 \dots g_k$ 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 M\"obius function of

length up to four.

functions, and let $h_0,\dots,h_k \in {\bf Z}$ be shifts. We consider

correlation sequences $f: {\bf N} \to {\bf Z}$ of the form $$ f(a):=

\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) \dots g_k(n+ah_k)}{n} $$ where $1 \leq \omega_m

\leq x_m$ are numbers going to infinity as $m \to \infty$, and

$\widetilde{\lim}$ is a generalised 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 $g_0 \dots g_k$ "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 $g_0 \dots g_k$ 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 M\"obius function of

length up to four.

## Symplectic id:

935445

## Download URL:

## Submitted to ORA:

Submitted

## Publication Type:

Journal Article