  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 $Q(x)\ge -\pi/2$ 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 $Q(x) < -\pi/2$. 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 $Q(x)<-\pi/2$.
