A logarithmic inequality involving prime numbers

Abstract

Assume that $N$ is a sufficiently large positive number. In this paper we show that for a small constant $\epsilon >0$, the logarithmic inequality \begin{equation*} \big|p_1\log p_1+p_2\log p_2+p_3\log p_3-N\big|<\varepsilon \end{equation*} has a solution in prime numbers $p_1,p_2,p_3$.