Riesz means of the Dedekind function II

TL;DR

This paper analyzes the Riesz means of the Dedekind totient function, providing explicit error term representations under the Riemann Hypothesis and Gonek-Hejhal Hypothesis, and proposing a proposition equivalent to the Riemann Hypothesis.

Contribution

It offers an explicit error term formula for the Riesz means of n/ψ(n) and improves error estimates assuming the Gonek-Hejhal Hypothesis, also proposing an equivalent Riemann Hypothesis condition.

Findings

Explicit representation of the error term in Riesz means of n/ψ(n).

Improved error estimates under the Gonek-Hejhal Hypothesis.

Proposed a proposition equivalent to the Riemann Hypothesis.

Abstract

Let $\psi$ denote the Dedekind totient function defined by $ \psi(n)=\sum_{d|n}d\mu^2\l({n}/{d}\r) $ with $\mu$ being the M\"{o}bius function. We shall consider the $k$-th Riesz mean of the arithmetical function $n/\psi(n)$ for any non-negative integer $k$ on the assumptions that the Riemann Hypothesis is true, and all the zeros $\rho$ on the critical line of the Riemann zeta function $\zeta$ are simple. Our result is an explicit representation of the error term in the formula obtained in a previous work of the second author and I. Kiuchi \cite{IK}. We also give an improvement on the error estimate under the assumption of the Gonek-Hejhal Hypothesis. And, we propose a proposition that is equivalent to the Riemann Hypothesis.