The distribution of $k$-free numbers,Mathematics of Computation

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The distribution of $k$-free numbers
Mathematics of Computation ( IF 2 ) Pub Date : 2020-11-24 , DOI: 10.1090/mcom/3581
Michael J. Mossinghoff , Tomás Oliveira e Silva , Timothy S. Trudgian

Let $R_k(x)$ denote the error incurred by approximating the number of $k$-free integers less than $x$ by $x/\zeta(k)$. It is well known that $R_k(x)=\Omega(x^{\frac{1}{2k}})$, and widely conjectured that $R_k(x)=O(x^{\frac{1}{2k}+\epsilon})$. 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

中文翻译：

$k$-free 数字的分布

让 $R_k(x)$ 表示用 $x/\zeta(k)$ 来逼近小于 $x$ 的 $k$-free 整数的数量所产生的误差。众所周知，$R_k(x)=\Omega(x^{\frac{1}{2k}})$，广泛推测$R_k(x)=O(x^{\frac{1}{1}{1}{2k}}) 2k}+\epsilon})$。通过建立黎曼 zeta 函数的一些零点子集的弱线性独立性，我们建立了下界的有效证明，与先前工作中获得的那些相比，常数的界明显更大。例如，我们证明 $R_k(x)/x^{1/2k} > 3$ 无限频繁并且 $R_k(x)/x^{1/2k} < -3$ 无限频繁，因为 $k= 2 美元、3 美元、4 美元和 5 美元。我们还详细研究了 $R_2(x)$ 和 $R_3(x)$ 并确定我们的界限远远超过这些函数在很长一段时间内表现出的振荡：对于 $0

更新日期：2020-11-24

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