Truncated convolution of Möbius function and multiplicative energy of an integer $n$

Abstract

We establish an interesting upper bound for the moments of truncated Dirichlet convolution of Möbius function, a function noted $M(n,z)$. Our result implies that $M(n,j)$ is usually quite small for $j\in \{1,\dots ,n\}$. Also, we establish an estimate for the multiplicative energy of the set of divisors of an integer $n$.