By modifying Beukers’ proof of Apéry’s theorem that
ζ
(
3
)
\zeta (3)
is irrational, we derive criteria for irrationality of Euler’s constant,
γ
\gamma
. For
n
>
0
n>0
, we define a double integral
I
n
I_n
and a positive integer
S
n
S_n
, and prove that with
d
n
=
LCM
(
1
,
…
,
n
)
d_n=\operatorname {LCM}(1,\dotsc ,n)
the following are equivalent: 1. The fractional part of
log
S
n
\log S_n
is given by
{
log
S
n
}
=
d
2
n
I
n
\{\log S_n\}=d_{2n}I_n
for some
n
n
. 2. The formula holds for all sufficiently large
n
n
. 3. Euler’s constant is a rational number. A corollary is that if
{
log
S
n
}
≥
2
−
n
\{\log S_n\}\ge 2^{-n}
infinitely often, then
γ
\gamma
is irrational. Indeed, if the inequality holds for a given
n
n
(we present numerical evidence for
1
≤
n
≤
2500
)
1\le n\le 2500)
and
γ
\gamma
is rational, then its denominator does not divide
d
2
n
(
2
n
n
)
d_{2n}\binom {2n}{n}
. We prove a new combinatorial identity in order to show that a certain linear form in logarithms is in fact
log
S
n
\log S_n
. A by-product is a rapidly converging asymptotic formula for
γ
\gamma
, used by P. Sebah to compute
γ
\gamma
correct to 18063 decimals.