Maximum drag reduction (MDR) in a turbulent channel flow by polymer additives is studied using direct numerical simulation. An Oldroyd-B model is adopted to express the polymer stress because MDR is closely related to the elasticity of the polymer solution. The Reynolds number considered is 4000, based on the bulk velocity and the channel height, and the amount of MDR from the present study is 44%, which is in good agreement with Virk's asymptote at this Reynolds number. For ‘large drag reduction’, the variations of turbulence statistics such as the mean streamwise velocity and r.m.s. velocity fluctuations are quite different from those of ‘small drag reduction’. For example, for small drag reduction, the r.m.s. streamwise velocity fluctuations decrease in the sublayer but increase in the buffer and log layers with increasing Weissenberg number, but they decrease in the whole channel for large drag reduction. As the flow approaches the MDR limit, the significant decrease in the production of turbulent kinetic energy is compensated by the increase in energy transfer from the polymer elastic energy to the turbulent kinetic energy. This is why turbulence inside the channel does not disappear but survives in the MDR state.