We investigate theoretically the possibility of using the cold-electron bolometer (CEB) as a counter for 1 cm wavelength (30 GHz) photons. To reduce the flux of photons from the environment, which interact with the detector, the bath temperature is assumed to be below 50 mK. At such temperatures, the time interval between two subsequent photons of 30 GHz that hit the detector is more than 100 hours, on average, for a frequency window of 1 MHz. Such temperatures allow the observation of the physically significant photons produced in rare events, like the axions conversion (or Primakoff conversion) in magnetic field. We present the general formalism for the detector's response and noise, together with numerical calculations for proper experimental setups. We observe that the current-biased regime is favorable, due to lower noise, and allows for the photons counting at least below 50 mK. For the experimental setups investigated here, the voltage-biased CEBs may also work as photons counters, but with less accuracy and eventually at bath temperatures below 40 mK. The voltage-biased setups also require smaller volumes of the normal metal island of the detector.

9 pages 13 figures, accepted in Phys. Rev. Allied, in production