%0 Journal Article
%T Strong orthogonality between the Mobius function and nonlinear exponential functions in short intervals
%A Bingrong Huang
%J Mathematics
%D 2014
%I arXiv
%R 10.1093/imrn/rnv091
%X Let $\mu(n)$ be the M\"obius function, $e(z) = \exp(2\pi iz)$, $x$ real and $2\leq y \leq x$. This paper proves two sequences $(\mu(n))$ and $(e(n^k \alpha))$ are strongly orthogonal in short intervals. That is, if $k \geq 3$ being fixed and $y\geq x^{1-1/4+\varepsilon}$, then for any $A>0$, we have \[ \sum_{x< n \leq x+y} \mu(n) e\left(n^k \alpha \right) \ll y(\log y)^{-A} \] uniformly for $\alpha \in \mathbb{R}$.
%U http://arxiv.org/abs/1412.2237v2