On $p$-adic Euler $L$-functions

Abstract

In this paper, we define the p-adic Euler L-functions using the fermionic p-adic integral on Zp. By computing the values of the p-adic Euler L-functions at negative integers, we show that for Dirichlet characters with odd conductor, this definition is quivalent to the previous definition following Kubata-Leopoldt and Washington's approach. We also study the behavior of p-adic Euler L-functions at positive integers. An interesting thing is that most of the results in Section 11.3.3 of Cohen's book [H. Cohen, Number Theory Vol. II: Analytic and Modern Tools, Graduate Texts in Mathematics, 240. Springer, New York, 2007] are also established if we replace the generalized Bernoulli numbers with the generalized Euler numbers. Comment: 17 pages

Full text

Page 1

arXiv:1010.4440v2 [math.NT] 29 Nov 2010
ON p-ADIC EULER L-FUNCTIONS
SU HU AND MIN-SOO KIM
Abstract. In this paper, we define the p-adic Euler L-functions us-
ing the fermionic p-adic integral on Zp. By computing the values of the
p-adic Euler L-functions at negative integers, we show that for Dirich-
let characters with odd conductor, this definition is equivalent to the
previous definition in [8] following Kubata-Leopoldt and Washington’s
approach. We also study the behavior of p-adic Euler L-functions at
positive integers. An interesting thing is that most of the results in Sec-
tion 11.3.3 of Cohen’s book [3] are also established if we replace the
generalized Bernoulli numbers with the generalized Euler numbers.
1. Introduction
Throughout this paper, we use the following notations.
C − the field of complex numbers.
p − an odd rational prime number.
Zp − the ring of p-adic integers.
Qp − the field of fractions of Zp.
Cp − the completion of a fixed algebraic closure Qpof Q.
CZp − Qp\Zp.
vp − the p-adic valuation of Cpnormalized so that |p|p= p−vp(p)= p−1.
Z×
p − the group of p-adic units.
In 1964, Kubota-Leopoldt [14] defined the p-adic analogues of classical
L-functions of Dirichlet which are closely related to the arithmetic of cy-
clotomic fields (see [5]). In [3, Chapter 11]), we can find another definition
for the p-adic L-functions using Volkenborn integrals which is equivalent to
Kubota-Leopoldt’s original definition. The Volkenborn integral was intro-
duced by Volkenborn [19] and he also investigated many important proper-
ties of p-adic valued functions defined on the p-adic domain (see [19, 20]).
In 1749, Euler gave a paper to the Berlin Academy entitled Remar-
ques sur un beau rapport entre les s´ eries des puissances tant directes que
r´ eciproques. In this paper, he studied
φ(s) =
∞
?
n=1
(−1)n
ns
2000 Mathematics Subject Classification. 11E95, 11B68, 11S80, 11M35.
Key words and phrases. Euler number and polynomial, p-adic integral, p-adic Euler
L-function.
1

Page 2

2SU HU AND MIN-SOO KIM
(see [1, p. 1081]).
For s ∈ C and Re(s) > 0, the Euler zeta function and the Hurwitz-type
Euler zeta function are defined by
ζE(s) = 2
∞
?
n=1
(−1)n
ns
, and ζE(s,x) = 2
∞
?
n=0
(−1)n
(n + x)s
respectively (see [1, 6, 11, 12, 15]). Notice that the Euler zeta functions
can be analytically continued to the whole complex plane, and these zeta
functions have the values of the Euler numbers or the Euler polynomials at
negative integers.
Recently, we defined the p-adic Hurwitz-type Euler zeta functions using
the fermionic p-adic integral on Zp(see [9]). Notice that the fermionic p-
adic integral on Zphas been used by T. Kim [11] to derive useful formulas
involving the Euler numbers and polynomials, and it has also been used by
the second author to give a brief proof of Stein’s classical result on Euler
numbers modulo power of two (see [7]). In [9], we first gave the definition
for the p-adic Hurwitz-type Euler zeta function for x ∈ CZp, then we gave
the definition for the p-adic Hurwitz-type Euler zeta function for x ∈ Zp
using characters modulo pvand also gave a new definition for the p-adic
Euler ℓ-function for characters modulo pvas a special case. We showed
that in this case the definition is equivalent to the second author’s previous
definition in [8]. In [8], the second author defined the p-adic Euler ℓ-functions
for Dirichlet characters with odd conductor following Kubota-Leopoldt’s
approach and Washington’s one and he also computed the derivative of p-
adic Euler ℓ-function at s = 0 and the values of p-adic Euler ℓ-function at
positive integers.
In this paper, we define the p-adic Euler L-functions using the fermionic
p-adic integral on Zp. By computing the values of the p-adic Euler L-
functions at negative integers, we show that for Dirichlet characters with
odd conductor, this definition is equivalent to the second author’s previ-
ous definition in [8] following Kubata-Leopoldt and Washington’s approach
(see Remark 4.6). We also study the behavior of p-adic Euler L-functions
at positive integers. We show that most of the results in Section 11.3.3 of
Cohen’s book [3] are also established if we replace the generalized Bernoulli
numbers with the generalized Euler numbers.
2. Generalized Euler numbers
In this section, we recall some facts on the Euler numbers, the Euler
polynomials and the generalized Euler numbers which will be used in the
subsequent sections to study the properties for the p-adic Euler L-functions.
The definition of Euler polynomials is well known and can be found in
many classical literatures (cf. see [17, p. 527-532]).
We consider the following generating function
(2.1)F(t,x) =
2ext
et+ 1.

Page 3

ON p-ADIC EULER L-FUNCTIONS3
Expand F(t,x) into a power series of t :
(2.2)F(t,x) =
∞
?
n=0
En(x)tn
n!.
The coefficients En(x),n ≥ 0, are called Euler polynomials. We note that
there is a second kind Euler numbers, namely, from (2.1)
(2.3)En= En(0),
For example
E0= 1,E1= −1
2,E3=1
4,E5= −1
2,E7=17
8,E9= −31
2,E11=691
4
,...
and E2k= 0 for k = 1,2,....
These numbers is a kind of (λ-)Euler numbers which defined by T. Kim
in [11, p. 784]. This is different from the first kind Euler numbers named by
Scherk in 1825 (see the first paragraph of [7, p. 2167]).
We generalize the above definition of Enand En(x) as follows: let χ be
a primitive Dirichlet character χ with an odd conductor f = fχ, and let
(2.4)Fχ(t) = 2
f
?
a=1
(−1)aχ(a)eat
eft+ 1
,|t| <π
f
and
(2.5)Fχ(t,x) = Fχ(t)ext= 2
f
?
a=1
(−1)aχ(a)e(a+x)t
eft+ 1
,|t| <π
f.
(see e.g. [11, p. 783]). Expanding theses into power series of t, let
(2.6)Fχ(t) =
∞
?
n=0
En,χtn
n!
and
(2.7)Fχ(t,x) =
∞
?
n=0
En,χ(x)tn
n!.
Then En,χ(x) =
En,χ(x) for n ≥ 0, are called generalized Euler numbers and polynomials,
respectively. It is clear from (2.4) that Fχ(−t) = −χ(−1)Fχ(t), if χ ?= χ0,
the trivial character. Hence we obtain the result as follows:
?n
i=0
?n
i
?Ei,χxn−ifor n ≥ 0. The coefficients En,χ and
Proposition 2.1. If χ ?= χ0, the trivial character, then we have
(−1)n+1En,χ= χ(−1)En,χ,n ≥ 0.
In particular we obtain
En,χ= 0if χ ?= χ0, n ?≡ δχ
(mod 2),
where δχ= 0 if χ(−1) = −1 and δχ= 1 if χ(−1) = 1.

Page 4

4SU HU AND MIN-SOO KIM
Let N be the least common multiple of p and f. Then by (2.4), we have
(2.8)Fχ(t) = 2
∞
?
m=0
(−1)mχ(m)emt= 2
N
?
a=1
(−1)aχ(a)(e
a
N)Nt
eNt+ 1.
Therefore, by (2.1), (2.2), (2.4), (2.6) and (2.8), we obtain the result as
follows:
Proposition 2.2. Let N be the least common multiple of p and f. Then
En,χ= Nn
N
?
a=1
(−1)aχ(a)En
?a
N
?
.
From (2.5), the generating function Fχ(t,x) is given by
(2.9)
Fχ(t,x) = 2
f
?
∞
?
a=0
(−1)aχ(a)
∞
?
k=0
(−1)ke(a+x+fk)t
=
n=0
?
2
∞
?
l=0
(−1)lχ(l)(l + x)n
?
tn
n!.
Comparing coefficients of tn/n! on both sides of (2.7) and (2.9) gives
(2.10)En,χ(x) = 2
∞
?
l=0
(−1)lχ(l)(l + x)n
(see [11, p.784]). In particular, if n ≥ 0 we have
(2.11)En,χ(x + N) = 2
∞
?
l=0
(−1)lχ(l)(l + x + N)n.
By (2.10) and (2.11), we obtain the results as follows:
Proposition 2.3. Let N be the least common multiple of f and p, we have
N−1
?
r=0
(−1)rχ(r)(x + r)n=En,χ(x) − (−1)NEn,χ(x + N)
2
.
3. The p-adic fermionic integral
In this section, we recall the p-adic fermionic integral and its relationship
between the Euler numbers and polynomials.
Let UD(Zp,Cp) denote the space of all uniformly (or strictly) differen-
tiable Cp-valued functions on Zp(see [11]). The metric space Qphas a basis
of open sets consisting of all sets of the form a+pNZp= {x ∈ Qp| |x−a|p≤
p−N} ⊂ Zpfor a ∈ Qpand N ∈ Z. This means that any open subset of
Qpis a union of open subsets of this type. For N ≥ 1, the p-adic fermionic
distribution
µ−1(a + pNZp) = (−1)a
(3.1)
is known as a measure on Zp(see [11]). We shall write dµ−1(x) to remind
ourselves that x is the variable of integration. It is well known that for

Page 5

ON p-ADIC EULER L-FUNCTIONS5
f ∈ UD(Zp,Cp), its fermionic p-adic integral I−1(f) on Zpis defined to be
the limit of
pN−1
?
as N → ∞ (see [11, 7, 9]). Write down this integral as
a=0
f(a)(−1)a
(3.2)I−1(f) =
?
Zp
f(x)dµ−1(x) = lim
N→∞
pN−1
?
a=0
f(a)(−1)a∈ Cp
(see [11] for more details). By (3.2), it is well known that the Euler numbers
En and the Euler polynomials En(x) are connected with I−1-integrals as
follows. For any n ≥ 0,
(3.3)I−1(xn) =
?
Zp
xndµ−1(x) = En
and
(3.4)I−1((x + y)n) =
?
Zp
(x + y)ndµ−1(y) = En(x)
(see [11, 7]).
We can now obtain the generalized Euler numbers and polynomials on
Zp. Let χ be a primitive Dirichlet character χ with an odd conductor f = fχ,
and let N be the least common multiple of p and f. Then by (3.2), (3.4)
and Proposition 2.2, we see that
(3.5)
?
Zp
χ(x)xndµ−1(x) = lim
r→∞
Npr−1
?
N−1
?
N−1
?
N−1
?
a=0
(−1)aχ(a)an
= lim
r→∞
k=0
pr−1
?
a=0
(−1)Na+kχ(Na + k)(Na + k)n
= Nn
k=0
(−1)kχ(k)
?
Zp
?k
?k
N+ a
?n
dµ−1(a)
= Nn
k=0
(−1)kχ(k)En
N
?
= En,χ
is equivalent to
(3.6)I−1(χ(x)xn) =
?
Zp
χ(x)xndµ−1(x) = En,χ.
Similarly, by (3.5) we have
(3.7)I−1(χ(y)(x + y)n) =
?
Zp
χ(y)(x + y)ndµ−1(y) = En,χ(x).
Therefore we have the results as follows:

Page 6

6SU HU AND MIN-SOO KIM
Proposition 3.1. Let χ be a primitive Dirichlet character χ with an odd
conductor. Then for n ≥ 0 we have
(1) I−1(xn) =?
(3) I−1(χ(x)xn) =?
Zpxndµ−1(x) = En.
Zp(x + y)ndµ−1(y) = En(x).
Zpχ(x)xndµ−1(x) = En,χ.
(4) I−1(χ(y)(x + y)n) =?
4. p-adic Euler L-functions
(2) I−1((x + y)n) =?
Zpχ(y)(x + y)ndµ−1(y) = En,χ(x).
In this section, we define the p-adic Euler L-functions and compute its
value at negative integers.
Given x ∈ Zp,p ∤ x and p > 2, there exists a unique (p − 1)-th root of
unity ω(x) ∈ Zpsuch that
x ≡ ω(x) (mod p),
where ω is the Teichm¨ uller character. Let ?x? = ω−1(x)x, so ?x? ≡ 1
(mod p). We extend the notation ? ? to Q×
pby setting
(4.1)?x? =
?
x
pvp(x)
?
.
If x ∈ Q×
p, we define ωv(x) by
(4.2)ωv(x) =
x
?x?= pvp(x)ω
?
x
pvp(x)
?
(see [3, p. 280, Definition 11.2.2] for more details). Now we give a definition
for the p-adic Euler L-functions.
Definition 4.1. Let χ be a primitive character of conductor f and m be
the least common multiple of f and p. For s ∈ Cpsuch that |s| < Rp=
p(p−2)/(p−1), we have
Lp,E(χ,s) = ?m?1−s
f−1
?
a=0
χ(a)ζp,E
?
s,a
f
?
(−1)a,
where the p-adic Hurwitz-type Euler zeta function ζp,E(s,x) is defined in
Definition 3.2 and 4.1 of [9] for x ∈ CZpand x ∈ Zp, respectively. We define
Lp,E(χ,1) = lim
trivial character χ0, we set ζp,E(s) = Lp,E(χ0,s) = ζp,E(s,0) (see Theorem
4.10 (3) in [9]).
s→1Lp,E(χ,s), when the limit exist. In particular if χ is the
Definition 4.2 (see [3, p. 302]).
be the trivial character modulo 1 when p ∤ m, and to be the trivial
character modulo p when p | m. In other words, χ0,m(a) = 1 when
p ∤ a or when p | a but p ∤ m and χ0,m(a) = 0 when p | a and p | m.
(2) If I ⊂ Z, we set
(1) Let m ∈ Z>0. We define χ0,m to
?(p)
a∈I
g(a) =
?
p∤a
a∈I
g(a)

Page 7

ON p-ADIC EULER L-FUNCTIONS7
and similarly
?(p)
a∈I
g(a) =
?
p∤a
a∈I
g(a).
In particular, if p | m we have
?(p)
0≤a<m
g(a) =
m−1
?
a=0
χ0,m(a)g(a).
Lemma 4.3 (see [3, p. 302]). Let χ be a nontrivial primitive character of
conductor f and m be a common multiple of f and p. Then
?(p)
0≤a<m
χ(a) = 0.
Proposition 4.4. Let χ be a primitive character of an odd conductor f
and m be the least common multiple of f and p. Let s ∈ Cpbe such that
|s| < Rp.
(1) We have
Lp,E(χ,s) = ?m?1−s
m−1
?
a=0
χ0,m(a)χ(a)ζp,E
?
s,a
m
?
(−1)a.
(2) If, in addition, p | m, we have
Lp,E(χ,s) = ?m?1−s?(p)
(3) If χ ?= χ0, then
0≤a<m
χ(a)(−1)a?a?1−s
∞
?
i=0
?1 − s
i
?mi
aiEi.
Lp,E(χ,1) =
?(p)
0≤a<m
χ(a)(−1)a.
Proof. (1) Writing a = kf + r, we have
(4.3)
?m?1−s
m−1
?
f−1
?
a=0
χ0,m(a)χ(a)ζp,E
?
s,a
m
?
(−1)a
= ?m?1−s
r=0
χ(r)(−1)r
m
f−1
?
k=0
χ0,m(kf + r)ζp,E
?
s,kf + r
m
?
(−1)k.
Case (I): If p ∤ m, then χ0,m to be a trivial character modulo 1 by
Definition 4.2. Using Theorem 3.10 (3) in [9], we have
m
f−1
?
k=0
χ0,m(kf + r)ζp,E
?
s,kf + r
m
?
(−1)k=
m
f−1
?
k=0
ζp,E
?
s,r
m+k
m
f
?
(−1)k
= ζp,E
?
s,r
f
?
.

Page 8

8SU HU AND MIN-SOO KIM
Case (II): If p | m, then χ0,m to be a trivial character modulo p by
Definition 4.2. Thus we have
m
f−1
?
k=0
χ0,m(kf + r)ζp,E
?
s,kf + r
m
?
(−1)k=
m
f−1
?
p∤(kf+r)
k=0
ζp,E
?
s,r
m+k
m
f
?
(−1)k.
Using Corollary 4.5 in [9], we obtain the following:
(a) Suppose p ∤ f. Then r/f ∈ Zpand
p ∤ (kf + r) if and only ifp ∤ (r/f + k).
Thus
(4.4)
m
f−1
?
p∤(kf+r)
k=0
ζp,E
?
s,r
m+k
m
f
?
(−1)k= ζp,E
?
s,r
f
?
.
(b) If p | f, then χ0,m(kf + r) = χ0,m(r). We have
m
f−1
?
k=0
χ0,m(kf + r)ζp,E
?
s,kf + r
m
?
(−1)k
= χ0,m(r)
m
f−1
?
k=0
ζp,E
?
s,r
m+k
m
f
?
(−1)k.
If p | r, then
(4.5)
m
f−1
?
k=0
χ0,m(kf + r)ζp,E
?
s,r
m+k
m
f
?
(−1)k= 0.
If p ∤ r, then r/f ∈ CZp. Thus by Theorem 3.10 (3) in [9] and the definition
of χ0,m(r), we have
(4.6)
χ0,m(r)
m
f−1
?
k=0
ζp,E
?
s,r
m+k
m
f
?
(−1)k= χ0,m(r)ζp,E
?
s,r
f
?
= ζp,E
?
s,r
f
?
.
Thus if p ∤ f, substitute (4.4) to (4.3), we have
?m?1−s
m−1
?
f−1
?
f−1
?
a=0
χ0,m(a)χ(a)ζp,E
?
s,a
m
?
(−1)a
= ?m?1−s
r=0
χ(r)(−1)r
m
f−1
?
k=0
χ0,m(kf + r)ζp,E
?
s,r
m+k
m
f
?
(−1)k
= ?m?1−s
r=0
χ(r)(−1)rζp,E
?
s,r
f
?
= Lp,E(χ,s).

Page 9

ON p-ADIC EULER L-FUNCTIONS9
If p | f, substitute (4.5) and (4.6) to (4.3), we have
?m?1−s
m−1
?
f−1
?
f−1
?
p∤r
a=0
χ0,m(a)χ(a)ζp,E
?
s,a
m
?
(−1)a
= ?m?1−s
r=0
χ(r)(−1)r
m
f−1
?
m
f−1
?
k=0
χ0,m(kf + r)ζp,E
?
s,r
m+k
m
f
?
(−1)k
= ?m?1−s
r=0
χ(r)(−1)r
k=0
χ0,m(kf + r)ζp,E
?
s,r
m+k
m
f
?
(−1)k
= ?m?1−s
f−1
?
p∤r
r=0
χ(r)(−1)rζp,E
?
s,r
f
?
= Lp,E(χ,s),
since p | f and p | r, we have (f,r) ?= 1, so χ(r) = 0, thus we get the last
equality.
(2) If p | m, from (1), we have
Lp,E(χ,s) = ?m?1−s
m−1
?
a=0
χ0,m(a)χ(a)ζp,E
?
s,a
m
?
(−1)a.
Since p | m, by Definition 4.2, we have
(4.7)
Lp,E(χ,s) = ?m?1−s
m−1
?
a=0
χ0,m(a)χ(a)ζp,E
?
s,a
m
?
(−1)a
= ?m?1−s?(p)
0≤a<m
χ(a)ζp,E
?
s,a
m
?
(−1)a.
Also since in the above equality, p ∤ a and p | m, we obtain
a
m∈ CZp.
By Theorem 3.5 of [9], we have
(4.8)ζp,E
?
s,a
m
?
=
?a
m
?1−s
∞
?
i=0
?1 − s
i
?
Eimi
ai,
where Eiare Euler numbers. Substitute (4.8) to (4.7), we have
Lp,E(χ,s) =
?(p)
0≤a<m
χ(a)(−1)a?a?1−s
∞
?
i=0
?1 − s
i
?mi
aiEi.
(3) Since p | m, by (2), we have
Lp,E(χ,1) =
?(p)
0≤a<m
χ(a)(−1)a
and the result is established.
?

Page 10

10SU HU AND MIN-SOO KIM
Next we compute the values of p-adic Euler L-functions at negative in-
tegers.
Proposition 4.5. Keep the above assumptions.
(1) The function Lp,E(χ,s) is a p-adic analytic function for |s| < Rp.
(2) For k ∈ Z≥1, we have
Lp,E(χ,1 − k) = (1 − pkχk(p))Ek,χk,
where χk= χω−kand χ ?= χ0, the trivial character.
(3) If χ is an even character the function Lp,E(χ,s) is identically equal
to zero
Remark 4.6. By comparing Proposition 4.5 (2) with equalities (3.2) and
(3.3) in [8, p. 6], from Lemma 1 in [5, p. 19] we conclude that for Dirichlet
characters with odd conductor the definition of p-adic Euler L-functions
in this paper is equivalent to the second author’s previous definition in [8]
following Kubata-Leopoldt’s approach.
Proof. (1) By definition of Lp,E(χ,s), Theorem 3.5 and Proposition 4.7 in
[9], the result follows.
(2) Let m be the least common multiple of p and f. Then by Proposition
4.4 (1), we have
Lp,E(χ,1 − k) = ?m?k
m−1
?
a=0
χ0,m(a)χ(a)ζp,E
?
1 − k,a
m
?
(−1)a.
Since p | m, by Definition 4.2 (2), we have
(4.9)
Lp,E(χ,1 − k) = ?m?k?(p)
Also since p ∤ a and p | m, we have a/m ∈ CZp. Using Theorem 3.9 (2) in
[9] we have
0≤a<m
χ(a)ζp,E
?
1 − k,a
m
?
(−1)a.
(4.10)
ζp,E
?
1 − k,a
m
?
=
1
?a
ωk
v
m
?Ek
?a
m
?
.
where Ek(x) are the Euler polynomials. Substitute (4.10) to (4.9), we have
Lp,E(χ,1 − k) = ?m?k?(p)
0≤a<m
χ(a)(−1)a
1
?a
v(a)Ek
ωk
v
m
?Ek
?a
m
?
= ?m?kωk
v(m)
?(p)
0≤a<m
(−1)aχω−k
?a
m
?
.
Since p ∤ a, we have
ωv(a) = pv(a)ω
?
a
pv(a)
?
= ω(a).

Page 11

ON p-ADIC EULER L-FUNCTIONS 11
Thus if χ ?= χ0, by Proposition 2.2, we have
Lp,E(χ,1 − k) = ?m?kωk
v(m)
?(p)
?m−1
a=0
0≤a<m
(−1)aχω−k(a)Ek
?a
?a
m
?
= ?m?kωk
v(m)
?
(−1)aχω−k(a)Ek
m
?
−
m−1
?
p|a
a=0
(−1)aχω−k
v(a)Ek
?a
m
??
=
?
mk
m−1
?
m
p−1
?
m−1
?
a=0
(−1)aχω−k(a)Ek
?a
m
?
− mk
a=0
(−1)aχω−k(pa)Ek
?pa
?a
m
??
=
?
mk
a=0
(−1)aχω−k(a)Ek
m
?
− pkχω−k(p)
?m
p
?k
m
p−1
?
a=0
(−1)aχω−k(a)Ek
?a
m
p
??
= (1 − pkχk(p))Ek,χk,
where χk= χω−k.
(3) From Definition 4.1, we have
Lp,E(χ,s) = ?m?1−s
f−1
?
a=0
χ(a)ζp,E
?
s,a
f
?
(−1)a.
Let b = f − 1 − a, and let χ be an even character. Then by Corollary 3.6
and Theorem 4.10 in [9], we have
Lp,E(χ,s) = ?m?1−s
f−1
?
f−1
?
f
?
b=0
χ(f − 1 − b)ζp,E
?
s,f − 1 − b
f
?
(−1)f−1−b
= ?m?1−s
b=0
χ(1 + b)ζp,E
?
s,1 + b
f
?
(−1)b
= ?m?1−s
b′=1
χ(b′)ζp,E
?
s,b′
f
?
(−1)b′(−1),
so Lp,E(χ,s) = −Lp,E(χ,s). Therefore Lp,E(χ,s) = 0 if χ is an even charac-
ter.
?

Page 12

12SU HU AND MIN-SOO KIM
5. p-adic Euler L-function at positive integers
In this section we study the behavior of p-adic Euler L-functions at pos-
itive integers following Cohen’s approach in Section 11.3.3 of [3]. We show
that most of the results in Section 11.3.3 of Cohen’s book are also estab-
lished if we replace the generalized Bernoulli numbers with the generalized
Euler numbers.
Proposition 5.1. Let f be an odd integer and χ be a primitive character
modulo f and m be the least common multiple of f and p.
(1) For k ∈ Z\{0}, we have
Lp,E(χ,k + 1) = lim
N→∞
?(p)
0≤n<mpN
χωk(n)(−1)n
nk
.
(2)
Lp,E(χ,1) = E0,χ,
where E0,χdefined in (2.6).
Proof. (1) By Theorem 3.9 (1) in [9], for p ∤ a, we have
(5.1)
ζp,E
?
k + 1,a
m
?
= ωk
v
?a
?a
?a
m
??
Zp
1
?a
m+ j?kdµ−1(j)
?
pN−1
?
pN−1
?
= ωk
v
m
?
lim
N→∞
pN−1
j=0
(−1)j
?a
m+ j?k
= ωk
v
m
?
mk
ωk
mklim
N→∞
j=0
(−1)j
(a + mj)k
= ωk
v(a)
v(m)
lim
N→∞
j=0
(−1)j
(a + mj)k.
So that, by Definition 4.2 (2) and Proposition 4.4 (1), we have
(5.2)
Lp,E(χ,k + 1) = ?m?−k
m−1
?
a=0
χ0,m(a)χ(a)ζp,E
?
k + 1,a
m
?
(−1)a
= ?m?−k?(p)
0≤a<m
χ(a)ζp,E
?
k + 1,a
m
?
(−1)a.
Thus if we set
n = mj + a, where 0 ≤ j ≤ pN− 1, 0 ≤ a ≤ m − 1, p ∤ a.
Then
0 ≤ n ≤ mpN− 1, and p ∤ m.

Page 13

ON p-ADIC EULER L-FUNCTIONS13
Substitute (5.1) to (5.2), since f is an odd integer and m is the least common
multiple of f and p, we have
(5.3)
Lp,E(χ,k + 1) =
?(p)
?(p)
?(p)
0≤a<m
χ(a)ωk
v(a)(−1)alim
N→∞
pN−1
?
(−1)j+a
(a + mj)k
j=0
(−1)j
(a + mj)k
=
0≤a<m
χ(a)ωk
v(a) lim
N→∞
pN−1
?
j=0
=
0≤a<m
χωk(a) lim
N→∞
pN−1
?
j=0
(−1)ma+j
(ma + j)k
= lim
N→∞
?(p)
0≤n<mpN−1
χωk(n)(−1)n
nk
.
(2) By Definition of Lp,E(χ,s), Theorem 3.8 and Corollary 4.4 in [9], we
have
Lp,E(χ,k + 1) =
f−1
?
f−1
?
a=0
χ(a)ζp,E
?
1,a
f
?
(−1)a
=
a=0
χ(a)(−1)a
= E0,χ.
?
Corollary 5.2.
power of p, then we have
Let k ∈ Z\{0}. If χ is a primitive character modulo a
Lp,E(χ,k + 1) =
?
Z×
p
χωk(x)
xk
dµ−1(x).
In particular,
Lp,E(ω−k,k + 1) =
?
Z×
p
1
xkdµ−1(x).
Proof. If f is a power of p, that is f = pv, then we have
Lp,E(χ,s) =
pv−1
?
a=0
χ(a)ζp,E
?
s,a
pv
?
(−1)a
= ζp,E(χ,s,0)
= ℓp,E(χ,s)

Page 14

14 SU HU AND MIN-SOO KIM
by Corollary 4.3 and Remark 4.11 in [9]. Thus
Lp,E(χ,s) = ℓp,E(x,k + 1)
= ζ(χ,s,0)
?
=
=
Zp
χ(x)?x?−kdµ−1(x)
?
?
?
Z×
p
χ(x)?x?−kdµ−1(x)
=
Z×
p
χωk(x)x−kdµ−1(x)
=
Z×
p
χωk(x)
xk
dµ−1(x).
?
Proposition 5.3. Let χ be a primitive character modulo f and Φ be the
Euler-phi function. Then for all k ∈ Z, we have
Lp,E(χ,k + 1) = lim
r→∞EΦ(pr)−k,χωk.
In particular,
lim
r→∞EΦ(pr)−k= Lp,E(ω−k,k + 1),
where Emare the Euler numbers.
Proof. Denote χk= χω−k. Since Lp,E(χ,s) is a continuous function of s, for
all k ∈ Z, we have
Lp,E(χ,k + 1) = lim
r→∞Lp,E(χ,k + 1 − Φ(pr))
= lim
r→∞Lp,E(χ,1 − (Φ(pr) − k)
= lim
r→∞(1 − pΦ(pr)−kχΦ(pr)−k(p))EΦ(pr)−k,χωk−Φ(pr)
using Proposition 4.5. Since
ωΦ(pr)= ωpr−1(p−1)= 1andχΦ(pr)−k= χωk−Φ(pr)= χωk,
thus
Lp,E(χ,k + 1) = lim
r→∞EΦ(pr)−k,χωk.
This completes the proof of the Theorem.
Definition 5.4. For k ∈ Z, we define the p-adic χ-Euler numbers by
?
Ek,p,χ= lim
r→∞EΦ(pr)+k,χ= Lp,E(χωk,1 − k).
Proposition 5.5. Assume that the conductor of χ is a power of p. Then
for k ∈ Z, we have
Ek,p,χ= lim
r→∞
?(p)
0≤n<pr
χ(n)nk(−1)n=
?
Zp
χ(x)xkdµ−1(x).

Page 15

ON p-ADIC EULER L-FUNCTIONS 15
Proof. We have
Ek,p,χ= Lp,E(χωk,1 − k)
?
= lim
r→∞
0≤n<pr
=
Z×
p
χ(x)xkdµ−1(x)
?(p)
χ(n)nk(−1)n.
using Corollary 5.2. Thus
Ek,p,χ= lim
r→∞
?(p)
0≤n<pr
χ(n)nk(−1)n=
?
Zp
χ(x)xkdµ−1(x).
?
Proposition 5.6.
(1) If χ(−1) = (−1)k, then we have
Ek,p,χ= 0.
(2) If k ≥ 1, then we have
Ek,p,χ= (1 − pkχ(p))Ek,χ.
(3) Let m be the least common multiple of f and p, and set
Hn(x) =
?(p)
0≤a<m
χ(a)
an(−1)a.
If k ≥ 1 and χ(−1) = (−1)k−1, we have
E−k,p,χ=
m
?
i=0
(−1)i
?k + i − 1
k − 1
?
miEiHk+i(x),
where Eiis the Euler numbers.
(4) For all k, we have vp(Ek,p,χ) ≥ 0.
Proof. (1) If χ(−1) = 1 and k ≡ 0 (mod 2), then Φ(pr) + k ≡ 0 (mod 2),
we have EΦ(pr)+k,χ= 0 by Proposition 2.1, thus
Ek,p,χ= lim
r→∞EΦ(pr)+k,χ= 0.
If χ(−1) = −1 and k ≡ 1( modulo 2), then Φ(pr)+k ≡ 1 (mod 2), we have
EΦ(pr)+k,χ= 0 by Proposition 2.1, thus
Ek,p,χ= lim
r→∞EΦ(pr)+k,χ= 0.
(2) By Proposition 4.5 (2) and Definition 5.4, we have
Ek,p,χ= Lp,E(χωk,1 − k) = (1 − pkχ(p))Ek,χ.