#### Vol. 11, No. 4, 2017

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Gowers norms of multiplicative functions in progressions on average

### Xuancheng Shao

Vol. 11 (2017), No. 4, 961–982
##### Abstract

Let $\mu$ be the Möbius function and let $k\ge 1$. We prove that the Gowers ${U}^{k}$-norm of $\mu$ restricted to progressions $\left\{n\le X:n\equiv {a}_{q}\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}q\right)\right\}$ is $o\left(1\right)$ on average over $q\le {X}^{1∕2-\sigma }$ for any $\sigma >0$, where ${a}_{q}\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}q\right)$ is an arbitrary residue class with $\left({a}_{q},q\right)=1$. This generalizes the Bombieri–Vinogradov inequality for $\mu$, which corresponds to the special case $k=1$.

##### Keywords
multiplicative functions, Bombieri–Vinogradov theorem, Gowers norms
##### Mathematical Subject Classification 2010
Primary: 11P32
Secondary: 11B30, 11N13