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
\[
G
(
x
)
≥
(
c
+
o
(
1
)
)
log
x
loglog
x
loglogloglog
x
(
logloglog
x
)
−
2
G(x) \geq (c + o(1)){\operatorname {log}}x{\operatorname {loglog}}x{\operatorname {loglogloglog}}x{({\operatorname {logloglog}}x)^{ - 2}}
\]
, where
c
=
e
γ
c = {e^\gamma }
and
γ
\gamma
is Euler’s constant. Here, this result is shown with
c
=
c
0
e
γ
c = {c_0}{e^\gamma }
where
c
0
=
1.31256
…
{c_0} = 1.31256 \ldots
is the solution of the equation
4
/
c
0
−
e
−
4
/
c
0
=
3
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.