# Truncated convolution of M\"obius function and multiplicative energy of an integer $n$

## TL;DR

This paper derives upper bounds for moments of a truncated Dirichlet convolution involving the Möbius function and estimates the multiplicative energy of an integer's divisors, revealing their typical smallness and structural properties.

## Contribution

It introduces new bounds for the moments of the truncated Möbius convolution and provides estimates for the multiplicative energy of divisor sets, advancing understanding of their behavior.

## Key findings

- $M(n,j)$ is usually quite small for $j \
- ,
- ,

## Abstract

We establish an interesting upper bound for the moments of truncated Dirichlet convolution of M\"obius 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 $n$.