Unusually large gaps between consecutive primes

Let G ( x ) G(x) denote the largest gap between consecutive primes below x x . In a series of papers from 1935 to 1963, Erdàs, Rankin, and Schànhage showed that \[ G ( x ) ≥ ( c + o ( 1 ) ) log x loglog x loglogloglog x ( logloglog x ) − 2 G(x) ≥ (c + o(1)){log}x{loglog}x{loglogloglog}x{({logloglog}x)- 2} \] , where c = e γ c = {e^γ } and γ γ is Euler’s constant. Here, this result is shown with c = c 0 e γ c = {c_0}{e^γ } where c 0 = 1.31256 … {c_0} = 1.31256 … is the solution of the equation 4 / c 0 − e −