  We extend the Collins-Soper-Sterman (CSS) formalism to apply it to the
spin-dependence governed by the Sivers function. We use it to give a correct
numerical QCD evolution of existing fixed-scale fits of the Sivers function.
With the aid of approximations useful for the non-perturbative region, we
present the results as parametrizations of a Gaussian form in transverse
momentum space, rather than in the Fourier conjugate transverse coordinate
space normally used in the CSS formalism. They are specifically valid at small
transverse momentum. Since evolution has been applied, our results can be used
to make predictions for Drell-Yan and semi-inclusive deep inelastic scattering
at energies different from those where the original fits were made. Our evolved
functions are of a form that they can be used in the same parton model
factorization formulas as used in the original fits, but now with a predicted
scale dependence in the fit parameters. We also present a method by which our
evolved functions can be corrected to allow for twist-3 contributions at large
parton transverse momentum.
