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

Abstract

We introduce a class of multiplicative functions in which each function satisfies some statistic conditions, and then prove that the 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 $G{L}_m(m⩾2)$ and polynomial nilsequences has logarithmic decay; the other is that the mean value of the Möbius function, coefficients of automorphic L-function, and polynomial nilsequences also has logarithmic decay.