Department of

November 2014 December 2014 January 2015 Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 1 2 3 4 5 6 1 2 3 2 3 4 5 6 7 8 7 8 9 10 11 12 13 4 5 6 7 8 9 10 9 10 11 12 13 14 15 14 15 16 17 18 19 20 11 12 13 14 15 16 17 16 17 18 19 20 21 22 21 22 23 24 25 26 27 18 19 20 21 22 23 24 23 24 25 26 27 28 29 28 29 30 31 25 26 27 28 29 30 31 30

Tuesday, December 2, 2014

**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 $\zeta(s)$. 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 $M_k(x)=\sum_{n\leq x}\mu_k(n)-\frac{x}{\zeta(k)}$ and $\mu_k(n)$ 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.