A method in connection with the production of mechanical pulp from a cellulose containing material wherein the material is processed in at least one refining step to produce pulp. According to the invention, the pulp is fractionated (4), after a first refining step (1), in order to separate primary fines (5) from the pulp. The invention also relates to a mechanical pulp, produced by the method, and to a paperboard, which at least partly consists of such mechanical pulp.