Hardy-Littlewood 函数的渐近逼近
Asymptotic approximations to the Hardy-Littlewood function
AI总结：
针对 Hardy-Littlewood 函数 Q(x) 在 x 很大时的逼近问题，本文发展了多种渐近逼近方法，并用于找到使 Q(x) < -π/2 的显式 x 值。
摘要：
函数 $Q(x):=\sum_{n\ge 1} (1/n) \sin(x/n)$ 由 Hardy 和 Littlewood [10] 在研究 Lambert 可和性时引入，此后引起了许多研究者的关注。特别是，该函数在 Alzer、Berg 和 Koumandos [1] 最近反驳 Clark 和 Ismail [2] 的一个猜想中出人意料地出现。更准确地说，Alzer 等人 [1] 已经证明，Clark 和 Ismail 猜想成立当且仅当对所有 $x>0$ 有 $Q(x)\ge -\pi/2$。已知 $Q(x)$ 在定义域 $x \in (0,\infty)$ 上无上界也无下界，这反驳了 Clark 和 Ismail 猜想，同时提出了一个自然的问题：我们能否至少找到一个点 $x$ 使得 $Q(x) < -\pi/2$。这被证明是一个出奇困难的问题，导致了一个有趣且非平凡的问题：如何对非常大的 $x$ 逼近 $Q(x)$。在本文中，我们继续 Gautschi 在 [7] 中开始的工作，发展了 $Q(x)$ 在 $x$ 很大时的几种逼近。我们利用这些逼近找到了一个显式的 $x$ 值，使得 $Q(x) < -\pi/2$。
英文摘要：
The function $Q(x):=\sum_{n\ge 1} (1/n) \sin(x/n)$ was introduced by Hardy and Littlewood [10] in their study of Lambert summability, and since then it has attracted attention of many researchers. In particular, this function has made a surprising appearance in the recent disproof by Alzer, Berg and Koumandos [1] of a conjecture by Clark and Ismail [2]. More precisely, Alzer et. al. [1] have shown that the Clark and Ismail conjecture is true if and only if $Q(x)\ge -π/2$ for all $x>0$. It is known that $Q(x)$ is unbounded in the domain $x \in (0,\infty)$ from above and below, which disproves the Clark and Ismail conjecture, and at the same time raises a natural question of whether we can exhibit at least one point $x$ for which $Q(x) < -π/2$. This turns out to be a surprisingly hard problem, which leads to an interesting and non-trivial question of how to approximate $Q(x)$ for very large values of $x$. In this paper we continue the work started by Gautschi in [7] and develop several approximations to $Q(x)$ for large values of $x$. We use these approximations to find an explicit value of $x$ for which $Q(x)<-π/2$.
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