This paper proposes a digital background calibration scheme for calibrating the linear and the third-order nonlinear gain errors of the residue amplifiers (RAs) in pipelined ADCs. It is based on a novel multiple-correlation estimation (MCE) technique. We define two correction parameters relating to the gain errors of the RA under calibration. By alternately injecting two bi-level pseudo-random signals with designated amplitudes to the RA through the sub-DAC, the desired correction parameters are estimated according to the correlations of the backend ADC's outputs and the injected pseudo-random signals. Two least-mean-square (LMS) loops are adopted to find and to track the optimal values of the correction parameters. Simulation results of a 12-bit pipelined ADC show that the SNDR is improved from 46.4 dB to 73.4 dB with the help of the proposed calibration design. The proposed calibration scheme has the advantages of simple implementation, no restriction on the input signal of the ADC, fast settling, and running in background.