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 .