On the Riesz means of ${\delta }_k(n)$

Abstract

Let $k\ge 1$ be an integer. Let ${\delta }_k(n)$ denote the maximum divisor of $n$ which is co-prime to $k$. We study the error term of the general $m$-th Riesz mean of the arithmetical function ${\delta }_k(n)$ for any positive integer $m\ge 1$, namely the error term $E_m(x)$ where \[ \frac{1}{m!}\sum_{n \leq x}δ_k(n) \left( 1-\frac{n}{x} \right)^m = M_{m, k}(x) + E_{m, k}(x). \] We establish a non-trivial upper bound for $\left|E_{m,k}(x)\right|$, for any integer $m\ge 1$.