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

Published online by Cambridge University Press:  08 July 2016

XIANCHANG MENG
XIANCHANG MENG*
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, U.S.A. e-mail: [email protected]
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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 $e^{-\frac{y}{2k}}M_k(e^y)$ and study the tail of the limiting distribution, where $M_k(x)=\sum_{n\leq x}\mu_k(n)-{x}/{\zeta(k)}$ and μk(n) is the characteristic function of k-free numbers. Finally, we make a conjecture about the maximum order of Mk(x) by heuristic analysis on the tail of the limiting distribution.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 162 , Issue 2 , March 2017 , pp. 293 - 317
Copyright
Copyright © Cambridge Philosophical Society 2016 

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