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One of the intriguing problems of modern geophysics is the occurrence of ionospheric precursors of earthquakes and their mechanisms. Many assumptions have been made about possible mechanisms of connection between tectonic processes and the state of the ionosphere. One of the hypotheses suggests that the ionosphere is affected by a vertical atmospheric current of seismogenic origin. This current can be caused by turbulent transport of charged aerosols in the lower atmosphere, ionization of the surface atmospheric layer due to pre-seismic growth of the radioactive radon, or other reasons. In this report, a theoretical analysis of this hypothesis is carried out under fairly general assumptions about the origin of this seismogenic current. The report presents a theoretical model of the stationary distribution of electric field in the atmosphere and ionosphere caused by a vertical current generated in the lower atmosphere. The atmospheric conductivity is assumed to increase exponentially with height, and the E-region of the ionosphere is considered in the approximation of a thin gyrotropic layer. An analytical solution of the problem has been obtained, which determines the distribution of currents and electric fields in all media for different current sources. The analysis shows that the electrical disturbances caused by this current rapidly decrease with height due to two main reasons: firstly, an exponential increase in the atmospheric conductivity with height and, secondly, the horizontal spreading of currents along the E-layer and the magnetically conjugate region of the ionosphere. The analytical solution of the problem made it possible to derive approximate formulas relating the amplitudes of E-field on the ground and in the ionosphere. The estimates of ratios of the near-surface and ionospheric electric fields were compared with the data of satellite and ground-based measurements of anomalous disturbances, which could presumably be associated with seismic events. On the basis of this comparison it was concluded that the hypothesis of the influence of the seismogenic current on the ionosphere looks unrealistic, regardless of the nature of the seismogenic current. Thus, it is necessary to examine in a qualitative way other physical mechanisms that can explain the effect of seismic events on the ionosphere. The study is supported by the grant 22-17-00125 from the Russian Science Foundation.