Asymptotic approximations to the Hardy-Littlewood function

作者：Kuznetsov A^*

来源：Journal of Computational and Applied Mathematics, 2013, 237(1): 603-613.

DOI：10.1016/j.cam.2012.06.038

• Hardy-Littlewood function
• Exponential sum
• Euler-Maclaurin formula
• Poisson summation

The function Q(x) := Sigma(n %26gt;= 1) (1/n) sin (x/n) was introduced by Hardy and Littlewood (1936) [5] in their study of Lambirt 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) %26gt;= -pi/2 for all x %26gt; 0. It is known that Q(x) is unbounded in the domain x is an element of (0, infinity) 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) %26lt; -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 (2005) in [4] 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) %26lt; -pi/2.

• 出版日期2013-1-1