The distribution of $k$-free numbers
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- by Michael J. Mossinghoff, Tomás Oliveira e Silva and Timothy S. Trudgian HTML | PDF
- Math. Comp. 90 (2021), 907-929 Request permission
Abstract:
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 $0<x\leq 10^{18}$ we show that $\vert R_2(x)\vert < 1.12543x^{1/4}$ and $\vert R_3(x)\vert < 1.27417x^{1/6}$. We also present some empirical results regarding gaps between square-free numbers and between cube-free numbers.References
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Additional Information
- Michael J. Mossinghoff
- Affiliation: Center for Communications Research, Princeton, New Jersey 08540
- MR Author ID: 630072
- ORCID: 0000-0002-7983-5427
- Email: m.mossinghoff@idaccr.org
- Tomás Oliveira e Silva
- Affiliation: Departamento de Electrónica, Telecomunicações e Informática / IEETA, Universidade de Aveiro, Portugal
- ORCID: 0000-0002-8878-3219
- Email: tos@ua.pt
- Timothy S. Trudgian
- Affiliation: School of Science, The University of New South Wales Canberra, Australia
- MR Author ID: 909247
- Email: t.trudgian@adfa.edu.au
- Received by editor(s): December 10, 2019
- Received by editor(s) in revised form: June 24, 2020
- Published electronically: November 24, 2020
- Additional Notes: This work was supported in part by a grant from the Simons Foundation (#426694 to the first author). The third author was supported by Australian Research Council Future Fellowship FT160100094
- © Copyright 2020 American Mathematical Society
- Journal: Math. Comp. 90 (2021), 907-929
- MSC (2020): Primary 11M26, 11N60; Secondary 11Y35
- DOI: https://doi.org/10.1090/mcom/3581
- MathSciNet review: 4194167