Higher uniformity of arithmetic functions in short intervals II. Almost all intervals

Abstract

We study higher uniformity properties of the von Mangoldt function Λ$$\Lambda $$, the Möbius function μ$$\mu $$, and the divisor functions dk$$d_{k}$$ on short intervals (x,x+H]$$(x,x+H]$$ for almost all x∈[X,2X]$$x \in [X, 2X]$$. Let Λ♯$$\Lambda ^{\sharp }$$ and dk♯$$d_{k}^{\sharp }$$ be suitable approximants of Λ$$\Lambda $$ and dk$$d_{k}$$, G/Γ$$G/\Gamma $$ a filtered nilmanifold, and F:G/Γ→C$$F \colon G/\Gamma \to \mathbb{C}$$ a Lipschitz function. Then our results imply for instance that when X1/3+ε≤H≤X$$X^{1/3+\varepsilon } \leq H \leq X$$ we have, for almost all x∈[X,2X]$$x \in [X, 2X]$$, supg∈Poly(Z→G)|∑x0$$A>0$$, and that when Xε≤H≤X$$X^{\varepsilon } \leq H \leq X$$ we have, for almost all x∈[X,2X]$$x \in [X, 2X]$$, supg∈Poly(Z→G)|∑x