Walter Gautschi

The Hardy-Littlewood Function: An Exercise in Slowly Convergent Series

The function in question is $H(x)=\sum_{k=1}^\infty \sin(x/k)/k$ --- a slowly convergent series. A summation procedure is first decribed 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 conjecture of C. Berg and H. Alzer, according to which H(x) is bounded from below by $-\pi/2$.