Proof: Suppose that . If then divides the numerator of both and . If and r is sufficiently large thenso that Therefore, letting and then , we obtain (10.1).
For each , definewhere 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 , 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 , one can also show thatwhere : note that has a pole at s=1.)
Proof: As whenever p does not divide j, we haveby 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).