Strong orthogonality between the M\"obius function, additive characters, and Fourier coefficients of cusp forms

TL;DR

This paper demonstrates strong oscillations in the argument of the product of Fourier coefficients of cusp forms, the Möbius function, and additive characters, revealing deep orthogonality properties in number theory.

Contribution

It establishes new orthogonality results between Fourier coefficients, the Möbius function, and additive characters for Hecke--Maass cusp forms, highlighting their strong oscillatory behavior.

Findings

Proves strong oscillations of the argument of the product involving Fourier coefficients and Möbius function.

Shows orthogonality between Fourier coefficients, Möbius function, and additive characters.

Reveals deep interactions and oscillatory nature in the structure of cusp form coefficients.

Abstract

Let $\nu_{f}(n)$ be the $n$-th nomalized Fourier coefficient of a Hecke--Maass cusp form $f$ for ${\rm SL}(2,\Z)$ and let $\alpha$ be a real number. We prove strong oscillations of the argument of $\nu_{f}(n)\mu (n) \exp (2\pi i n \alpha)$ as $n$ takes consecutive integral values.