We give explicit upper bounds for the Stirling numbers of the first kind $s(n,m)$ which are asymptotically sharp. The form of such bounds varies according to m lying in the central or non-central regions of $\{1,\dots ,n\}$. In each case, we use a different probabilistic representation of $s(n,m)$ in terms of well known random variables to show the corresponding upper bounds. Some applications concerning the Riemann zeta function and a certain subset of the Comtet numbers of the first kind are also provided.

中文翻译：

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第一类斯特林数的明确上界

我们给出了第一类斯特林数的明确上限$s(n,米)$它们是渐近尖锐的。这种边界的形式根据位于中心或非中心区域的m而变化$\{1,\dots ,n\}$. 在每种情况下，我们使用不同的概率表示$s(n,米)$用众所周知的随机变量来显示相应的上限。还提供了有关黎曼 zeta 函数和第一类 Comtet 数的某个子集的一些应用。