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 $\{n\le X:n\equiv a_q(\mathrm{mod}q)\}$ is $o(1)$ on average over $q\le X^{1/2-\sigma }$ for any $\sigma >0$, where $a_q(\mathrm{mod}q)$ is an arbitrary residue class with $(a_q,q)=1$. This generalizes the Bombieri-Vinogradov inequality for $\mu$, which corresponds to the special case $k=1$.

20 pages

References in corpus (2)

• An averaged form of Chowla's conjecture
• Bombieri--Vinogradov and Barban--Davenport--Halberstam type theorems for smooth numbers

Cited by in corpus (2)

• Bombieri-Vinogradov for multiplicative functions, and beyond the $x^{1/2}$-barrier
• The Bombieri-Vinogradov theorem for nilsequences