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In section 11 we will apply the results from this section to binomial coefficients.
We start by proving

** Proposition 3.** * If ***x** is divisible by prime **p** then
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).