Top panels: SMBH mass function between redshift 0 and 4 as predicted by the models with different choices of the radiative efficiency and the local mass function. At z < 1, the BHMF evolves only at M < 10⁸M_⊙, while number density of very massive SMBHs undergoes a strong evolution a higher redshifts. At z > 1, details of the SMBH cosmic history strongly depend on the radiative efficiency. Central panels: duty cycle U(M,z) as predicted by the models. At z < 1, the duty cycle is ~ 1 for SMBHs of M < 10⁷M_⊙ and then decreases with the mass. The behavior of the duty cycle is very similar for the models with ϵ = 0.1 and 0.07, increasing from redshift 0 to 1, and decreasing at z > 2. For ϵ = 0.05, it always increases and most SMBHs are active at z ~ 4.Bottom panels: AGN MF as predicted by the models. This is almost insensitive to the choice of the local mass function and to the value of the radiative efficiency. The number density of active SMBHs with M ≥ 10⁸M_⊙ peaks at redshifts 1–2 and then rapidly decreases.