The Hardy-Littlewood function: an exercise in slowly convergent series
Two procedures for computing the values of the Hardy-Littlewood function \[ H(x)=\sum_{k=1}^{\infty}\frac{1}{k}\sin \frac{x}{k} \] are presented. The first one is based on Gaussian quadrature, while the second method uses direct summation for the first \(n\) terms of the series, where \(n\approx x\), in combination with some well-known formulae for zeta functions. Such calculations are of interest for exploring the possible boundedness of \(H\) from above and from below.
scientific article; zbMATH DE number 2173197