On prime numbers of special kind on short intervals

Suppose that the Riemann hypothesis holds. Suppose that $$psi _1 (x) = mathop sum limits_{begin{array}{*{20}c} {n leqslant x} {{ (1/2)n^{1/c} } < 1/2} end{array} } Lambda (n)$$ where c is a real number, 1 < c ≤ 2. We prove that, for H>N ^1/2+10ε, ε > 0, the following asymptotic formula is valid: $$psi _1 (N + H) - psi _1 (N) = frac{H}{2}left( {1 + Oleft( {frac{1}{{N^varepsilon }}} right)} right)$$ .