On prime numbers of special kind on short intervals

Suppose that the Riemann hypothesis holds. Suppose that
$$ \psi_1(x)=\sum_{\substack{n\le x\\ \{(1/2)n^{1/c}\}<1/2}}\Lambda(n), $$
where $c$ is a real number, $1<c\le 2$. We prove that, for $H>N^{1/2+10\varepsilon}$, $\varepsilon>0$, the following asymptotic formula is valid:
$$ \psi_1(N+H)-\psi_1(N)=\frac H2\biggl(1+O\biggl(\frac1{N^\varepsilon}\biggr)\biggr). $$