Discorrelation of multiplicative functions with nilsequences and its application on coefficients of automorphic <i>L</i> ‐functions

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