Let $p_n$ be the $n$th prime. Then this paper is concerned withproving the following result on the distribution of consecutive primes.

Theorem.}\begin{equation}\sum_{p_{n+1}-p_n>x^{\frac 12},\ x \leq p_n \leq 2x} (p_{n+1}-p_n)\llx^{\frac{25}{36}+\epsilon}.\end{equation}

The exponent of $x$ in this theorem improves on the workof Heath-Brown who proved $(1)$ with exponent $\frac 34$. Under the Riemann hypothesis one can prove$(1)$ withexponent $\frac 12$.The proof of the theorem startswith the Heath-Brown--Linnik identity which leads to aformula giving the number of primes in an interval in terms of coefficients of certain Dirichlet series. Ithen estimate the coefficients by using, among other things, the information which can be gained fromMontgomery's mean value theorem and Huxley's version of the Hal\' asz lemma. Furthermore, by using familiarsieve arguments I am able to discard some of the coefficients allowing us to gain an improvement over theprevious result of Heath-Brown.

1991 Mathematics Subject Classification: 11N05.