Asymptotic approximations to the Hardy-Littlewood function

TL;DR

This paper develops asymptotic approximations for the Hardy-Littlewood function to identify specific points where it falls below a critical threshold, addressing a longstanding open problem in the function's behavior.

Contribution

It introduces new asymptotic methods to approximate Q(x) for large x, enabling the explicit identification of points where Q(x) < -π/2, advancing understanding of the function's unboundedness.

Findings

Established asymptotic approximations for large x

Identified explicit x where Q(x) < -π/2

Confirmed unboundedness of Q(x) in both directions

Abstract

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