Given a smooth, irreducible complex algebraic variety
X
X
and a nonzero regular function
f
f
on
X
X
, we give an effective estimate for the difference between the jumping numbers of
f
f
and the
F
F
-jumping numbers of a reduction
f
p
f_p
of
f
f
to characteristic
p
≫
0
p\gg 0
, in terms of the roots of the Bernstein-Sato polynomial
b
f
b_f
of
f
f
. In particular, we get uniform estimates only depending on the dimension of
X
X
. As an application, we show that if
b
f
b_f
has no roots of the form
−
l
c
t
(
f
)
−
n
-lct(f)-n
, with
n
n
a positive integer, then the
F
F
-pure threshold of
f
p
f_p
is equal to the log canonical threshold of
f
f
for
p
≫
0
p\gg 0
with
(
p
−
1
)
l
c
t
(
f
)
∈
Z
(p-1)lct(f)\in {\mathbf Z}
.