The author proves that, at a point
P
P
on a closed Riemann surface of genus
g
g
, if
h
h
is the first nongap at
P
P
and
k
k
is relatively prime to
h
h
, then
k
k
is a gap if
g
>
1
2
(
h
−
1
)
(
k
−
1
)
g > \tfrac {1}{2}(h - 1)(k - 1)
. A consequence is that at the Weierstrass points of a closed Riemann surface, if the first nongap is a prime, the situation mirrors that in the hyperelliptic case, at least in a limiting sense.