The distribution of k-free numbers

2019/12/10 by Mossinghoff, Michael J., Silva, Tomás Oliveira e, Trudgian, Tim
#11M26 #11N60 #11Y35 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1912.04972

Abstract

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^(1)/(2k)), and widely conjectured that R_k(x)=O(x^(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 R₂(x) and R₃(x) in detail and establish that our bounds far exceed the oscillations exhibited by these functions over a long range: for 0

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