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

TL;DR

This paper establishes strong orthogonality between the Möbius function and nonlinear exponential functions over short intervals, extending understanding of their independence in analytic number theory.

Contribution

It proves that the Möbius function and nonlinear exponential sequences are strongly orthogonal in short intervals for fixed powers k ≥ 3, with explicit bounds.

Findings

Orthogonality holds for y ≥ x^{1 - 1/4 + ε}

Sum of Möbius times exponential is bounded by y (log y)^{-A}

Results are uniform for all real α

Abstract

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 $\alpha \in \mathbb{R}$.