The Hardy-Littlewood function: an exercise in slowly convergent series

The function in question is H(x)=∑k=1∞sin(x/k)/k. In deference to the general theme of this conference, a summation procedure is first described using orthogonal polynomials and polynomial/rational Gauss quadrature. Its effectiveness is limited to relatively small (positive) values of x. Direct summation with acceleration is shown to be more powerful for very large values of x. Such values are required to explore a (in the meantime disproved) conjecture of Alzer and Berg, according to which H(x) is bounded from below by -π/2.