Strong orthogonality between the Mobius function and nonlinear exponential functions in short intervals

Abstract

Let $\mu (n)$ be the Möbius function, $e(z)=\exp (2\pi iz)$, $x$ real and $2\le y\le x$. This paper proves two sequences $(\mu (n))$ and $(e(n^k\alpha ))$ are strongly orthogonal in short intervals. That is, if $k\ge 3$ being fixed and $y\ge x^{1-1/4+\epsilon }$, then for any $A>0$, we have \[ \sum_{x< n \leq x+y} μ(n) e\left(n^k α\right) \ll y(\log y)^{-A} \] uniformly for $\alpha \in \mathbb{R}$.

21 pages. Comments are welcome, Int Math Res Notices (2015)

References in corpus (1)

• The mean-square of Dirichlet L-functions