Gowers norms of multiplicative functions in progressions on average

Abstract

Let $\mu$ be the M\"{o}bius function and let $k \geq 1$. We prove that the Gowers $U^k$-norm of $\mu$ restricted to progressions $\{n \leq X: n\equiv a_q\pmod{q}\}$ is $o(1)$ on average over $q\leq X^{1/2-\sigma}$ for any $\sigma > 0$, where $a_q\pmod{q}$ is an arbitrary residue class with $(a_q,q) = 1$. This generalizes the Bombieri-Vinogradov inequality for $\mu$, which corresponds to the special case $k=1$.Comment: 20 page

Bombieri-Vinogradov for multiplicative functions, and beyond the x1/2-barrier

“…Note that these bounds are non-trivial since the number of x ε -smooth integers up to x is ≫ x. Combining Corollary 1.7 with the machinery developed in [29], one may prove for such multiplicative functions that their higher Gowers U k -norms are o(1) in progressions on average. This result will be stated and discussed in Section 9.…”

“…Breaking the x 1/2 -barrier. The main method used in our proofs is a modification of that developed by Green in [21]; see also [29] for using a similar argument to deal with higher Gowers norms. Green proved (a more general result which implies) that…”

“…Proof. In view of Corollary 1.7, the hypothesis on f implies that More precisely, if q ≤ Q is an exceptional moduli in the sense that f (q · +a q ) U k (x/q) ≥ ε for some residue class a q (mod q), where 0 ≤ a q < q and (a q , q) = 1, then by the inverse theorem for Gowers norms [22] f must correlate with a nilsequence of complexity O Corollary 9.1 only applies to smoothly supported f since we need to save an arbitrary power of log x in the Bombieri-Vinogradov inequality for f for the results in [29] to be applicable. However we would guess that our estimate max 0≤a<q (a,q)=1 f (q · +a) U k (x/q) = o(1) holds on average over q ≤ x 1/2−o(1) , for any 1-bounded multiplicative function f satisfying the 1-Siegel-Walfisz criterion.…”

The Bombieri-Vinogradov theorem for nilsequences

“…The lemma there is formulated for 1-bounded completely multiplicative functions, but the same proof works for any 1-bounded function f , as long as we do not require the condition (a , q ) = 1 in that lemma. So we can apply [29,Lemma 2.4] to the function f q / log x (whose L ∞ -norm is bounded).…”

“…Since 20D 2 ≤ M and since ψ d is totally δ C -equidistributed for large enough C, we may apply [29,Lemma 3.3] to bound the double sum above by O(δ 4 N 2 M 2 ) = O(δ 4 x 2 ). The conclusion follows immediately.…”