Unusually large gaps between consecutive primes

Abstract:

Let $G(x)$ denote the largest gap between consecutive primes below $x$. In a series of papers from 1935 to 1963, Erdàs, Rankin, and Schànhage showed that \[ G(x) \geq (c + o(1)){\operatorname {log}}x{\operatorname {loglog}}x{\operatorname {loglogloglog}}x{({\operatorname {logloglog}}x)^{ - 2}}\], where $c = {e^\gamma }$ and $\gamma$ is Euler’s constant. Here, this result is shown with $c = {c_0}{e^\gamma }$ where ${c_0} = 1.31256 \ldots$ is the solution of the equation $4/{c_0} - {e^{ - 4/{c_0}}} = 3$. 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.