Let
C
C
be the Casimir operator on a compact, simple, simply connected Lie group
G
G
of dimension
n
n
. The number of eigenvalues of
C
C
, counted with their multiplicities, of absolute value less than or equal to
t
t
is asymptotic to
k
t
n
/
2
k t^{n/2}
,
k
k
a constant. This paper shows the error of this estimate to be
O
(
t
2
b
+
a
(
a
−
1
)
/
(
a
+
1
)
)
O({t^{2b + a(a - 1)/(a + 1)}})
; where
a
a
= rank of
G
G
and
b
=
1
/
2
(
n
−
a
)
b = 1/2 (n - a)
.