On the asymptotic behaviour of the integral integral(infinity)(0) e(itx) (1/x(alpha)-1/(x(alpha))+1) dx (t -> 0) and rates of convergence to alpha-stable limit laws

We study the behaviour of the exponential sums Sigma (n greater than or equ al to1) exp{it n(1/alpha)}/n(n+1) in a small neighbourhood of t = 0 for 0 < <alpha> less than or equal to 1. Our main result yields an exact estimate of the remainder term in the corresponding Tauberian theorem. In particular , we prove that [GRAPHICS] as t --> 0. These asymptotic relations provide optimal uniform rates of con vergence in limit theorems for partial sums of independent random variables with common distribution function F-alpha(x) = 1 - 1/[x(alpha)]+1 for x > 0.