The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures
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 $$ f(a):= \widetilde{\lim}_{m \to \infty} \frac{1}{\log \omega_m} \sum_{x_m/\omega_m \leq n \leq x_m...
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| Format: | Journal article |
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Duke University Press
2019
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| _version_ | 1797060155914846208 |
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| author | Tao, T Teräväinen, J |
| author_facet | Tao, T Teräväinen, J |
| author_sort | Tao, T |
| collection | OXFORD |
| description | Let $g_0,\dots,g_k: {\bf N} \to {\bf D}$ be $1$-bounded multiplicative functions, and let $h_0,\dots,h_k \in {\bf Z}$ be shifts. We consider correlation sequences $f: {\bf N} \to {\bf Z}$ of the form $$ 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. |
| first_indexed | 2024-03-06T20:13:26Z |
| format | Journal article |
| id | oxford-uuid:2b5a3d31-9e0d-45c6-8b80-0a3e80f47ec3 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T20:13:26Z |
| publishDate | 2019 |
| publisher | Duke University Press |
| record_format | dspace |
| spelling | oxford-uuid:2b5a3d31-9e0d-45c6-8b80-0a3e80f47ec32022-03-26T12:30:24ZThe structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjecturesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:2b5a3d31-9e0d-45c6-8b80-0a3e80f47ec3Symplectic Elements at OxfordDuke University Press2019Tao, TTeräväinen, JLet $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. |
| spellingShingle | Tao, T Teräväinen, J The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures |
| title | The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures |
| title_full | The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures |
| title_fullStr | The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures |
| title_full_unstemmed | The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures |
| title_short | The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures |
| title_sort | structure of logarithmically averaged correlations of multiplicative functions with applications to the chowla and elliott conjectures |
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