where the limit is taken p--adically.
Proof: Suppose that 
. If 
then
divides the numerator of both 
and 
. If 
and r is sufficiently large then
so that
Therefore, letting 
and then 
, we obtain (10.1).
For each 
, define
where the limit here is taken p-adically: Note that this limit
exists and is well defined by (8.2); moreover 
for all odd n.
(Using Theorem 5.11 of [21], one can also show that
,
where 
is the p--adic L--function, and 
is that
p--adic 
st root of unity for which
.)
Our main result of this section is
Proposition 4. For any integer x we have
where 
(
) is chosen so that
, and
Remark: Note that 
if 
, and
if p=2.
(Using Theorem 5.11 of [21], one can also show that
where 
: note that 
has a pole at s=1.)
Proof: As 
whenever
p does not divide j, we have
by Proposition 3. Now, fix 
and take 
with r
large in (8.3), so
that
for all sufficiently large r, as
for those k in the sum.
Therefore, letting 
and then 
, we obtain (10.2).
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