On M\"obius functions from automorphic forms and a generalized Sarnak's conjecture

TL;DR

This paper proves that certain M"obius functions related to automorphic forms are orthogonal to bounded sequences, supporting a generalized Sarnak's conjecture for these functions.

Contribution

It establishes the weak orthogonality of M"obius functions from specific automorphic L-functions to bounded sequences, extending Sarnak's conjecture.

Findings

M"obius functions from Rankin-Selberg L-functions are orthogonal to bounded sequences

M"obius functions from Maass cusp forms are orthogonal to bounded sequences

Generalized Sarnak's conjecture holds for these automorphic M"obius functions

Abstract

In this paper, we consider M\"obius functions associated with two types of $L$-functions: Rankin-Selberg $L$-functions of symmetric powers of distinct holomorphic cusp forms and $L$-functions of Maass cusp forms. We show that these M\"obius functions are weakly orthogonal to bounded sequences. As a direct corollary, a generalized Sarnak's conjecture holds for these two types of M\"obius functions.