Abstract: On the base of the Mertens's first theorem, the more accurate estimates in the Mertens' first theorem were improved by using the idea of the integral estimates and the Stirling's formula. The improved Mertens's estimating inequalities were obtained. The results for the improved estimates were more accurate than the Mertens's first theorem in estimates. The improved estimates of Merterns's first theorem were applied to a specific asymptotic formula, in which the precise range of constant in Landau notation was increased.