The distribution of $k$-free numbers and the derivative of the Riemann zeta-function

TL;DR

This paper explores the connection between the distribution of $k$-free numbers and the derivative of the Riemann zeta-function at its zeros, under the Riemann Hypothesis, and investigates the limiting distribution of a related error term.

Contribution

It establishes a link between $k$-free numbers and zeta-function derivatives, proves the existence of a limiting distribution for a scaled error term, and conjectures its maximum order.

Findings

Existence of a limiting distribution for $e^{-rac{y}{2k}}M_k(e^y)$

Connection between $k$-free numbers and zeta zeros under RH

Heuristic conjecture on the maximum order of $M_k(x)$

Abstract

Under the Riemann Hypothesis, we connect the distribution of $k$-free numbers with the derivative of the Riemann zeta-function at nontrivial zeros of $\zeta(s)$. Moreover, with additional assumptions, we prove the existence of a limiting distribution of $e^{-\frac{y}{2k}}M_k(e^y)$ and study the tail of the limiting distribution, where $M_k(x)=\sum_{n\leq x}\mu_k(n)-\frac{x}{\zeta(k)}$ and $\mu_k(n)$ is the characteristic function of $k$-free numbers. Finally, we make a conjecture about the maximum order of $M_k(x)$ by heuristic analysis on the tail of the limiting distribution.