The best-known achievable rate region for the two-user Gaussian interference channel corresponds to the Han-Kobayashi scheme. However, mathematical expressions that characterize the Han-Kobayashi rate region are complicated. This complexity hinders a comprehensive understanding of the rate region. For instance, when interference is weak, the maximum achievable sum-rate of the Han-Kobayashi scheme has been unknown. This paper studies the sum-rate of the Han-Kobayashi scheme with Gaussian inputs and fully characterizes the maximum achievable sum-rate, when no time sharing is used. The optimal power-splitting variables and the corresponding maximum achievable sum-rate are explicitly expressed in closed forms. With the same approach, the maximum weighted sum-rate is expressed that characterizes the boundary of the Han-Kobayashi region without time sharing. Moreover, when time sharing is used, the boundary is expressed in terms of the upper concave envelope of a function of transmitters' powers.