We continue to study the distribution of prime numbers p, satisfying the condition $\{p^{\alpha} \} \in I \subset [0; 1)$, in arithmetic progressions. In this paper, we prove an analogue of Bombieri–Vinogradov theorem for 0 < α < 1/9 with the level of distribution $\theta = 2/5 - (3/5) \alpha$, which improves the previous result corresponding to $\theta \leqslant 1/3$.

The topic of the paper is an approach to find optimal regimes of miscible gas injection into the reservoir to maximize cumulative oil production using a surrogate model. The sector simulation model of the real reservoir with a gas cap, which is in the first stage of development, was used as a basic model for surrogate model training. As the variable (control) parameters of the surrogate model parameters of gas injection into injection wells and the limitation of the gas factor of production wells were chosen. The target variable is the dynamics of oil production from the reservoir. A set of data has been created to train the surrogate model with various input parameters generated by the Latin hypercube. Several machine learning models were tested on the data set: ARMA, SARIMAX and Random Forest. The Random Forest model showed the best match with simulation results. Based on this model, the task of gas injection optimization was solved in order to achieve maximum oil production for a given period. The optimization issue was solved by Monte Carlo method. The time to find the optimum based on the Random Forest model was 100 times shorter than it took to solve this problem using a simulator. The optimal solution was tested on a commercial simulator and it was found that the results between the surrogate model and the simulator differed by less than 9%.

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We continue to study the distribution of prime numbers p, satisfying the condition {p α } ∈ I ⊂ [0; 1), in arithmetic progressions. In the paper, we prove an analogue of Bombieri-Vinogradov theorem for 0 < α < 1/9 with the level of distribution θ = 2/5 − (3/5)α, which improves the previous result corresponding to θ 1/3.

Let α > 0 be any fixed non-integer, I be any subinterval of [0; 1). In the paper, we prove an analogue of Bombieri-Vinogradov theorem for the set of primes p satisfying the condition {p α } ∈ I. This strengthens the previous result of Gritsenko and Zinchenko.