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We study the properties of the maximum entropy spectral estimator in pure frequencies estimation. We model this situation by a sum of pure frequencies added to white noise. We study the effect of the signal-to-noise ratio, of the autoregressive filter order, and of the pure frequencies amplitude ratio on the resolving power. Neglecting the uncertainties in the autoregressive filter coefficient estimates, we show that the poles of the autoregressive filter transfer function can be classified into two families: one gives the pure frequencies and the other the white noise. By a first-order development we can specify the position of the pure frequencies associated poles. This allows us to give analytical results on the bias of the pure frequency estimation and on the resolving power. These theoretical results are confirmed and illustrated by computer simulations.