Exceptional characters and prime numbers in short intervals

Under the assumption of the Riemann hypothesis the asymptotic value y/log x is known to hold for the number of primes in the short interval [x - y, x] for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y = x^\alpha $$ \end{document} for every fixed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\alpha < {1\over 2}$$ \end{document}. We show under the assumption of the existence of exceptional Dirichlet characters the same asymptotic formula holds in the shorter intervals, for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\alpha < {1\over 2}$$ \end{document} \, in wide ranges of x depending on the characters.