The ultimate resolving power of the atomic frequency discriminator based on ESR of optically pumped potassium atoms was studied in quest of the best version of the Zeeman Earth-field magnetometer. A perfectly resolved single line as narrow as 1 Hz was realized at a signal-to-noise ratio not less than 10⁴ within the passband from 0.001 to 1 Hz, providing the ultimate short-term sensitivity better than 10 fT/Hz^1/2. Laser pumping provides ∼2 times better ultimate short-term resolution and a number of other technical advantages. The main source of systematic errors was found to be light-induced resonance frequency shifts, which were studied under laser pumping versus wavelength. The base-line stability of the stationary installed magnetometer is expected to be of about 10 pT under lamp excitation and can be made much better in the case of properly wavelength adjusted laser pumping. Although expected tilt errors up to 0.1 nT cannot be circumvented by laser pumping, they can be greatly suppressed in the gradiometric mode of magnetometers application when two or more identical sensors are mounted on a common rigid frame.