  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.
