Bombieri–Vinogradov Theorem in Shorter Intervals

Let \( y = x^{\theta} \) and \( Q = x^{\psi}(\log x)^{-B} \) where \( B = B(A) \). Using a recent large value estimate for Dirichlet L-functions proved by Chen, the author proves that $$ \sum_{q \leqslant Q} \max_{(a,q)=1} \max_{h \leqslant y} \max_{\frac{x}{2} \leqslant z \leqslant x} \left| \pi(z+h; q, a) - \pi(z; q, a) - \frac{\text{Li}(z+h) - \text{Li}(z)}{\varphi(q)} \right| \ll \frac{y}{(\log x)^{A}} $$ holds true for \( \theta > \frac{4}{7} \) and \( \psi < 2 \theta - \frac{8}{7} \). The ``interval length'' \( x^{\frac{4}{7} + \varepsilon} \) is shorter than any previous results of this type.