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Currently displaying 1 – 7 of 7

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On the distribution of $p^{α}$ modulo one

Xiaodong Cao; Wenguang Zhai — 1999

Journal de théorie des nombres de Bordeaux

In this paper, we give a new upper-bound for the discrepancy $D (N) : = sup_{0 \leq γ \leq 0} | \sum_{\binom{p \leq / N}{\{p^{α}\} \leq γ}} 1 - π (N) γ |$ for the sequence $(p^{α})$ , when $5 / 3 \leq α > 3$ and $α \neq 2$ .

The distribution of square-free numbers of the form $[n^{c}]$

Xiaodong Cao; Wenguang Zhai — 1998

Journal de théorie des nombres de Bordeaux

It is proved that the sequence $[n^{c}] (n = 1, 2, \dots)$ contains infinite squarefree integers whenever $1 < c < \frac{61}{36} = 1.6944 \dots$ , which improves Rieger’s earlier range $1 < c < 1.5$ .