Exponential sums over primes in short intervals

In this paper we establish one new estimate on exponential sums over primes in short intervals. As an application of this result, we sharpen Hua’s result by proving that each sufficiently large integer N congruent to 5 modulo 24 can be written as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$N = p_1^2 + p_2^2 + p_3^2 + p_4^2 + p_5^2 , with |p_j - \sqrt {N/5} | \leqslant U = N^{\frac{1}{2} - \frac{1}{{20}} + \varepsilon } $$ \end{document}, where pj are primes. This result is as good as what one can obtain from the generalized Riemann hypothesis.