Gowers norms of multiplicative functions in progressions on average

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 $\left\{n\le X:n\equiv a_q\left(modq\right)\right\}$ is $o\left(1\right)$ on average over $q\le X^{1∕2-\sigma }$ for any $\sigma >0$, where $a_q\left(modq\right)$ 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 $k=1$.