Unusually Large Gaps Between Consecutive Primes

Abstract

Let G(x) denote the largest gap between consecutive grimes below x, In a series of papers from 1935 to 1963, Erdos, Rankin, and Schonhage showed that G(x) :::: (c + o( I)) logx loglogx log log log 10gx(loglog logx) -2, where c = e Y and y is Euler's constant.Here, this result is shown with c = coe Y where Co = 1.31256... is the solution of the equation 4/ Co -e -4/co = 3 .The principal new tool used is a result of independent interest, namely, a mean value theorem for generalized twin primes lying in a residue class with a large modulus.