## Lectures on p-adic L-functions by Kinkichi Iwasawa

By Kinkichi Iwasawa

Those are notes of lectures given at Princeton college in the course of the fall semester of 1969. The notes current an advent to p-adic L-functions originated in Kubota-Leopoldt {10} as p-adic analogues of classical L-functions of Dirichlet.

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Hence The function L p (s; X) is uniquely characterized also by i) and by the above equalities. §3. l) -1, =0 If n '" 0 mod e, then n is even so that Bn, X � 0 by Theorem 2, §2. Hence, by the uniqueness mentioned above, if XC-I) = -1. On the other hand, if XC-I) 1, then for n == 0 mod e so that for n;:: 1, n == 0 mod e. Hence Lp Cs; X) is certainly not identically O. 5. converse of Theorem 1 is also partially true. Although this will never be used in the following (exce pt for an elementary lemma below), the result seems inter� esting enough to be mentioned here.

Ai d�i ) . i=o If A(x) beloogs to Q K ' it follows from Lemma 5, §3 that x

We have to sh ow that ! Yn(s)! ::: l n f ! for all s integer, m � 0, such that p-1 I m , Then n yn(m) "" 2: i=o n = (_1)n -i (i) ¢(i, m) I (-1 l-i (n im i=o pri < Z ' Let m be an p Zp 46 p -ADIC L-FUNCTIONS because ¢(i, m) = *m = ·m 1 fo r p Y i , p-1 1 m , and p fo. 2. It follows that n yn(m) '" l (_l)n -i i= o (�) im The sum on the right is the integer d�) introduced in 3 . 5 and we know (Lemma 5, §3) that it is divisible by n L Hence we have Now, the integers m such as mention ed above are everywhere dense in Zp ' Therefore lyn (s)1 s: In! *