The distribution of $k$-free numbers

Let $R_k(x)$ denote the error incurred by approximating the number of $k$-free integers less than $x$ by $x/ζ(k)$. It is well known that $R_k(x)=Ω(x^{\frac{1}{2k}})$, and widely conjectured that $R_k(x)=O(x^{\frac{1}{2k}+ε})$. By establishing weak linear independence of some subsets of zeros of the Riemann zeta function, we establish an effective proof of the lower bound, with significantly larger bounds on the constant compared to those obtained in prior work. For example, we show that $R_k(x)/x^{1/2k} > 3$ infinitely often and that $R_k(x)/x^{1/2k} < -3$ infinitely often, for $k=2$, $3$, $4$, and $5$. We also investigate $R_2(x)$ and $R_3(x)$ in detail and establish that our bounds far exceed the oscillations exhibited by these functions over a long range: for $0<x\leq10^{18}$ we show that $|R_2(x)| < 1.12543x^{1/4}$ and $|R_3(x)| < 1.27417x^{1/6}$. We also present some empirical results regarding gaps between square-free numbers and between cube-free numbers.

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