Asymptotic approximations to the Hardy–Littlewood function

The function Q(x):=∑n≥1(1/n)sin(x/n)Q(x):=∑n≥1(1/n)sin(x/n) was introduced by Hardy and Littlewood (1936) [5] 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 (2005) [3] of a conjecture by Clark and Ismail (2003) [14]. More precisely, Alzer et al. have shown that the Clark and Ismail conjecture is true if and only if Q(x)≥−π/2Q(x)≥−π/2 for all x>0x>0. It is known that Q(x)Q(x) is unbounded in the domain x∈(0,∞)x∈(0,∞) 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 xx for which Q(x)<−π/2Q(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)Q(x) for very large values of xx. In this paper we continue the work started by Gautschi (2005) in [4] and develop several approximations to Q(x)Q(x) for large values of xx. We use these approximations to find an explicit value of xx for which Q(x)<−π/2Q(x)<−π/2.