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The author gives nontrivial upper and lower bounds for the number of primesin the interval $[x - x^{\theta}, x]$ for some $0.52 \leqslant \theta < 0.525$,showing that the interval $[x - x^{0.52}, x]$ contains prime numbers for allsufficiently large $x$. This refines a result of Baker, Harman and Pintz (2001)and gives an affirmative answer to Harman and Pintz's argument. Newarithmetical information, a delicate sieve decomposition, various techniques inHarman's sieve and accurate estimates for integrals are used to good effect.

The number of primes in short intervals and numerical calculations for Harman's sieve