Some asymptotic formulas related to the Möbius function of higher order

Abstract In 1970, Apostol introduced the Möbius function of order k for an integer $k\ge 1$ . In 2001, it was generalized by Bege to the Möbius function $\mu _{k,m}$ of two parameters for any integers $m\ge k\ge 1$ . In this article, we will establish three kinds of asymptotic formulas related to $\mu _{k,m}$ for $m\ge k\ge 2$ in a unified elementary method. These formulas are related to the shifted convolution sums of Fourier coefficients of holomorphic cusp forms, Bergelson and Richter’s dynamical generalization of the prime number theorem, and the Titchmarsh divisor problem.