The Differences Between Consecutive Primes

This paper discusses two problems which relate to the difference between consecu~ve primes. Let p. be the nth prime, and d. = p . + l p . . (1) We want to find a funct ion f(,~) such that d . 0 is a given constant and cl is a number which satisfies ~*((r, T) ~O, (4) provided the funct ion g(x) satisfies g(x) K loglog x fox* every K > 0 and x>fzo(tC). This paper establishes the following theorem and so improves on the above result. Theorem 1. Suppose that the estimate (4) hoZds and the est~mata N (or, T) 0 and q'z>0. For any A > 0 ~e$