On two Diophantine inequalities over primes (II)

Let $1<c<\frac{26088036}{12301745},c≠2$ and $N$ be a sufficiently large real number. In this paper, it is proved that, for almost all $R\in (N,2N]$, the Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c-R\big|<\log^{-1}N \end{equation*} is solvable in primes $p_1,p_2,p_3$. Moreover, we also prove that the following Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c-N\big|<\log^{-1}N \end{equation*} is solvable in prime variables $p_1,p_2,p_3,p_4,p_5,p_6$, which improves the previous result $1<c<\frac{37}{18},c\mathrm{\ne }2$.