Stechkin's works in number theory

This paper is devoted to the analysis of S. B. Stechkin's contribution to some questions in analytic number theory. There are five areas of his research in the field of number theory. First, the works of S. B. Stechkin on the theory of the Riemann zeta function are considered. His results on even trigonometric polynomials played a role in these studies. Another area of research to which S. B. Stechkin made a significant contribution together with A. Y. Popov, relates to the asymptotic distribution of prime numbers on average. The third question, to which one of the works of S. B. Stechkin in analytic number theory was devoted, is related to Vinogradov’s mean value theorem, the main method for estimating Weyl sums. The fourth area of research, where S. B. Stechkin managed to get a result that could not be strengthened over the past 30 years, is the estimation of complete rational trigonometric sums. Finally, the fifth direction is the study of Gauss sums. Stechkin’s result in this direction and the problem he posed inspired followers to the present time.