On the Riesz means of $\delta_k(n)$

TL;DR

This paper investigates the error term in the Riesz mean of the arithmetic function elta_k(n), providing new upper bounds for the error for any positive integer order m, advancing understanding of divisor functions.

Contribution

It introduces novel upper bounds for the error term in the Riesz mean of elta_k(n), applicable for all positive integer orders m, extending previous results.

Findings

Established non-trivial upper bounds for the error term E_{m,k}(x).

Extended analysis to all positive integer m in the Riesz mean context.

Enhanced understanding of divisor-related arithmetic functions.

Abstract

Let $k\geq 1$ 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 \[ \frac{1}{m!}\sum_{n \leq x}\delta_k(n) \left( 1-\frac{n}{x} \right)^m = M_{m, k}(x) + E_{m, k}(x). \] We establish a non-trivial upper bound for $\left | E_{m, k} (x) \right |$, for any integer $m\geq 1$.