Discorrelation of multiplicative functions with nilsequences and its application on coefficients of automorphic $L$-functions

TL;DR

This paper proves that certain multiplicative functions are uncorrelated with polynomial nilsequences and applies this to show decay properties of automorphic L-function coefficients and the Möbius function.

Contribution

It introduces a class of multiplicative functions with specific statistical properties and demonstrates their lack of correlation with polynomial nilsequences, leading to new decay results.

Findings

Multiplicative functions with certain statistics are uncorrelated with polynomial nilsequences.

Twisting automorphic L-function coefficients with nilsequences exhibits logarithmic decay.

Mean values of the Möbius function and automorphic L-function coefficients decay logarithmically when paired with nilsequences.

Abstract

We introduce a class of multiplicative functions in which each function satisfies some statistic conditions, and then prove that above functions are not correlated with finite degree polynomial nilsequences. Besides, we give two applications of this result. One is that the twisting of coefficients of automorphic $L$-function on $GL_m (m \geq 2)$ and polynomial nilsequences has logarithmic decay; the other is that the mean value of the M\"obius function, coefficients of automorphic $L$-function and polynomial nilsequences also has logarithmic decay.