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

2014/08/02 by Xianchang Meng, Meng, Xianchang · 1 citation
Mathematics · #11M26 #11N56 #11N60 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1408.0429

openalex publication_date 2014/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ζ(s). Moreover, with additional assumptions, we prove the existence of a limiting distribution of e^-(y)/(2k)M_k(e^y) and study the tail of the limiting distribution, where M_k(x)=∑_{n≤ x}μ_k(n)-(x)/(ζ(k)) and μ_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.

Citations

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