A tangent inequality over primes

Abstract

In this paper we introduce a new diophantine inequality with prime numbers. Let $1<c<\frac{10}{9}$. We show that for any fixed $\theta >1$, every sufficiently large positive number $N$ and a small constant $\epsilon >0$, the tangent inequality \begin{equation*} \big|p^c_1\tan^θ(\log p_1)+ p^c_2\tan^θ(\log p_2)+ p^c_3\tan^θ(\log p_3) -N\big|<\varepsilon \end{equation*} has a solution in prime numbers $p_1,p_2,p_3$.