Table Of ContentA ZETA FUNCTION FOR MULTICOMPLEX ALGEBRA
6 A.SEBBAR,D.C.STRUPPA,A.VAJIAC,ANDM.B.VAJIAC
1
0
2 Abstract. In this paper we define and study a Dedekind-like zeta function
for thealgebra ofmulticomplex numbers. Byusingthe idempotent represen-
n tations for such numbers, we are able to identify this zeta function with the
a one associated to a product of copies of the field of Gaussian rationals. The
J
approach weuseis substantiallydifferent fromthe onepreviouslyintroduced
9 by Rochon (for the bicomplex case) and by Reid and Van Gorder (for the
1 multicomplexcase).
]
T
N
1. Introduction
.
h
In this paper we build on the resurgent interest for the theory of bicomplex
t
a and multicomplex numbers, to develop a definition (and discuss the fundamental
m
properties)ofaDedekind-likezetafunctionforthespacesofmulticomplexnumbers.
[ Our approach is significantly different from the one recently used by Rochon [19],
andReidandVanGorder[17]. Weshouldnotethatzetafunctionsplayasignificant
1
v role in a variety of fields, ranging from number theory to statistical mechanics,
5 from quantum field theory (where they are used to regularize divergent series and
8 divergent integrals) to dynamical systems, and finally to the theory of crystals
7
and quasi-crystals (see e.g. [3]). We believe that zeta functions for multicomplex
4
algebras will play an important role in a similar range of applications. From a
0
. mathematicalpoint ofview, we observethatthe study of the case ofmulticomplex
1
algebrasrepresentsonly afirststeptowardsthe understandingofthe seminalwork
0
of Hey [9] and Artin [2], within the larger context of quotient polynomialalgebras.
6
1 We plan to return to these issues in future papers.
: Tobeginwith, andwithoutpretenseofcompleteness,werecallthatthe spaceof
v
i bicomplex numbers arises when considering the space C of complex numbers as a
X
realbidimensionalalgebra,andthencomplexifyingit. Withthisprocessoneobtains
r a four dimensional algebra usually denoted by BC. The key point of the theory of
a
functions on this algebra is that (despite the problems posed by the existence of
zero-divisors in BC) the classical notion of holomorphicity can be extended from
one complex variable to this algebra, and one can therefore develop a new theory
of (hyper)holomorphic functions. Modern references on this topic are [1], and [13].
The algebra BC is therefore four dimensional over the reals, just like the skew-
field of quaternions, but while in the space of quaternions we have three anti-
commutative imaginary units, in the case of bicomplex numbers one considers two
imaginary units i,j which commute, and so the third unit k = ij ends up being a
“new” root of 1; such units are usually called hyperbolic. Indeed, every bicomplex
Date:January20,2016.
2010 Mathematics Subject Classification. 30G35,32A30,11M06,11R42.
Key words and phrases. bicomplex numbers, multicomplex algebra, Dedekind zeta function,
imaginaryquadraticfields.
1
2 A.SEBBAR,D.C.STRUPPA,A.VAJIAC,ANDM.B.VAJIAC
number Z can be written as Z = z +jz , where z and z are complex numbers
1 2 1 2
of the form z = x +iy , and z = x +iy . There are several ways to repre-
1 1 1 2 2 2
sent bicomplex numbers, see [13], but the one that will be central in this paper is
called the idempotent representation of bicomplex numbers, and will be described
in section 4.
One can then use a similar process to define the space BC of multicomplex
n
numbers, namely the space generated over the reals by n commuting imaginary
units. When n = 2, the space of multicomplex numbers is simply the space of
bicomplex numbers. The history of bicomplex numbers is not devoid of interest,
and we refer the reader to the recent [5].
In [19], the author introduced and studied the properties of a holomorphic Rie-
mann zeta function of two complex variables in the context of the bicomplex alge-
bra. Similarly, in [17] the authors defined a multicomplex Riemann zeta function
in the setup of multicomplex algebras. Both these studies generalize the Riemann
zeta function to severalcomplex variables,in the sense thatin the definition ofthe
original Riemann zeta function,
∞ 1
ζ(s):= ,
ns
n=1
X
thecomplexvariablesisreplacedbyabicomplex,respectivelyamulticomplexvari-
able. Our approach, in this paper, is very different. As it is well known, Dedekind
generalizedthe Riemannzeta functionbyconsideringanalgebraicnumberfieldK,
and defining the associated Dedekind zeta function by
1
ζ (s):=
K N(I)s
I⊂XOK
where the sum ranges through all the non-zero ideals I in the ring of integers
K
O
of K, and (see Section 3 for the full detail) N(I) denotes the norm of the ideal I.
When K =Q, the Dedekind zeta function reduces to the Riemann zeta function.
Thus, it is natural to look at the Dedekind approach for quadratic fields, and
concurrently the Hey [9] approach for hypercomplex algebras, as a way to define
a Dedekind-like zeta function in the context of the bicomplex and multicomplex
vector spaces BQ, respectively BQ . As the reader will see, the crucial point in
n
being able to explicitly calculate this type of zeta function (sometimes called the
Hey zeta function in the literature) for multicomplex numbers is the existence
of the idempotent representations of bicomplex and multicomplex numbers. This
representationwillallowustoidentifytheHeyzetafunctionsforproductsofcopies
of Q(i), the field of Gaussian rationals, and as a result we will have an explicit
formula for such a function.
The architecture of the paper is as follows. Section 2 gives a self-contained
review of quadratic fields, so that all necessary results are available to the reader.
In particular, we will define the quadratic L-function of a quadratic field K, and
we will calculate the value of its analytic extension for s = 1. Section 3 provides
allthe necessarybackgroundonthe Dedekind’s zeta function. InSection4 wegive
all the necessary information on bicomplex and multicomplex algebras. The core
of the paper is Section 5 where we utilize the instruments introduced previously
to calculate explicitly the Dedekind-like (Hey) zeta function for the algebra BQ
of bicomplex numbers with rational coefficients. We finally show how this results
ZETA FUNCTION FOR MULTICOMPLEX ALGEBRA 3
extends to the multicomplex case, and we explicitly write the functional equation
satisfied by these functions.
Acknowledgments. We wouldliketo thankAurelPageforfruitful discussions
about the topic of this paper.
2. Review on Quadratic Fields
Forthe convenienceofthe readerandinorderto makethe paperself-contained,
we summarize the main definitions and results for quadratic fields. Notations,
definitions, and results here follow [10, 14] and the references therein.
A number field is a finite degree field extension K over Q. We denote by
K
O
the ring of integers of K, i.e. the ring of elements α K that are roots of monic
∈
polynomials in Z[X].
A quadratic field is a degree two extension of Q. It has the form K = Q(√d),
where d is a squarefree integer different than 1. The ring of integers in this case is
Z+Z√d, if d 1 mod 4
6≡
Od :=OQ(√d) = Z+Z1+√d, if d 1 mod 4
2 ≡
The norm of an element α = a+b√d Q(√d) is the (non-necessarily positive)
∈
integer defined by
N(α):=(a+b√d)(a b√d)=a2 db2.
− −
The norm of an ideal I of is defined by
K
O
N(I):= /I ,
K
|O |
where we note that the quotient ring /I is always of finite cardinality for each
K
O
number field K. A particular case occurs when I =(α) is a principal ideal, where
α=a+b√d. Then
N(I)=N((α))= a2 db2 .
| − |
For an odd prime p Z, the Legendre symbol is defined by:
∈
p−1
a 1, if a 2 ≡1 mod p
p := −1, if ap−21 ≡−1 mod p
(cid:18) (cid:19) 0, if a 0 mod p.
≡
Recall that to say that a is aquadratic residue modulo p means that the equation
x2 = a mod p has a solution. We can therefore reformulate the definition of
Legendre symbol as follows:
1, if a is a quadratic residue modulo p and a 0 mod p
a 6≡
:= 1, if a is not a quadratic residue modulo p
p −
(cid:18) (cid:19) 0, if a 0 mod p.
≡
An extension of the Legendre symbol, due to Kronecker, is the following. Each
integer n has a prime factorization
n=upℓ1 pℓk,
1 ··· k
4 A.SEBBAR,D.C.STRUPPA,A.VAJIAC,ANDM.B.VAJIAC
where u= 1 and each p ,1 i k is prime. The Kronecker-Legendresymbol is
i
± ≤ ≤
defined by:
a a k a ℓi
:= ,
n u p
(cid:16) (cid:17) (cid:16) (cid:17)Yi=1(cid:18) i(cid:19)
where:
a
(1) for odd prime p, is the Legendre symbol;
p
(cid:18) (cid:19)
(2) for p=2, we define
0, if a is even
a
:= 1, if a 1 mod 8
p ≡±
(cid:18) (cid:19) 1, if a 3 mod 8.
− ≡±
(3) for u= 1 we define
±
a a 1, if a 0
=1, = ≥
1 1 1, if a<0.
(cid:16) (cid:17) (cid:18)− (cid:19) (cid:26) −
(4) we define
a 1, if a= 1
= ±
0 0, otherwise.
(cid:16) (cid:17) (cid:26)
AfundamentalresultforourstudyoftheDedekindzetafunctionisthefollowing
Theorem 2.1. Every non-zero ideal of can be written as a product of prime
d
O
ideals. The decomposition is unique up to the order of the factors.
The discriminant of the quadratic number field K =Q(√d) is defined by
d if d 1 mod 4
∆=∆K = 4d if d≡2,3 mod 4.
(cid:26) ≡
Some primes in Z are not prime elements in : for example, in the case d= 1,
d
O −
we have:
2=i(1 i)2,
−
5=(2+i)(2 i).
−
This situation is described precisely by the Legendre symbol. If p is a rational
prime (i.e. prime in Z), then the ideal (p)=p of has the following form:
d d
O O
∆
pp(where p=p), if =1
′ ′
6 p
(cid:18) (cid:19)
(p)= p, if (cid:18)∆p(cid:19)=−1
∆
wherep,p′ areprime idealspo2f, d. We respectiveilfy(cid:18)saypt(cid:19)ha=ti0n, these casesthe ideal
O
(p) splits, stays inert, or ramifies in .
d
O
ZETA FUNCTION FOR MULTICOMPLEX ALGEBRA 5
A subset F of is called a fractional ideal of if there exists β , β =0,
d d d
O O ∈O 6
such that βF is an ideal of . Then we have, for some ideal I of ,
d d
O O
α
F = α I ,
β ∈
(cid:26) (cid:12) (cid:27)
(cid:12)
The set FQ(√d) of fractional ideals of Od(cid:12)(cid:12) can be equipped with an abelian group
structure, as follows. If I ,I are ideals of , and
1 2 d
O
α α
1 2
F = α I F = α I ,
1 1 1 2 2 2
β ∈ β ∈
(cid:26) 1 (cid:12) (cid:27) (cid:26) 2 (cid:12) (cid:27)
(cid:12) (cid:12)
where β1,β2 d, we define:(cid:12) (cid:12)
∈O (cid:12) (cid:12)
α
F F := α I I
1 2 1 2
β β ∈
(cid:26) 1 2 (cid:12) (cid:27)
(cid:12)
where I1I2 is the ideal generated by all prod(cid:12)ucts α1α2, with α1 I1, α2 I2.
(cid:12) ∈ ∈
The identity element is and the inverse of a fractional ideal F is given by
d
O
F 1 = α Q(√d) αF .
− d
∈ ⊂O
(cid:26) (cid:12) (cid:27)
(cid:12)
The setof principalfractionalideals BQ(√d) ⊂(cid:12)(cid:12) FQ(√d) is a subgroupofFQ(√d), and
the quotient
H :=F B
d Q(√d) Q(√d)
is calledthe ideal class group of Q(√d). Its(cid:14)order h is the class number of Q(√d).
d
This notion measures how far the ring is from being principal.
d
O
Moreprecisely,ifh =1thenthereisonlyoneequivalenceclassinH ,andeach
d d
fractional ideal is equivalent to the principal ideal (1)= modulo multiplication
d
O
by principal ideals. That is, for each fractional ideal A there exists α such
d
∈ O
that
(α)=αA=(1)= ,
d
O
then
1
A= .
α
(cid:18) (cid:19)
Hence each fractional ideal and, therefore, each ideal is principal.
Thereexistnineimaginaryquadraticfieldsofclassnumberone. TheyareQ(√d)
for
d= 1, 2, 3, 7, 11, 19, 43, 67, 163.
− − − − − − − − −
If K =Q(√d), d<0, all norms a2 db2 are non-negative and the unit groupin
−
is
d
O
1, i if d= 1
{± ± } −
Od× = {±1,±ζ3,±ζ32} if d=−3
1 otherwise ,
{± }
where ζ3 is the principal cubic rootof unity. Therefore, the order of the group Od×
of units is:
w =w =4,6,2
d
according to the values d= 1, d= 3, or d= 1, 3, respectively.
− − 6 − −
6 A.SEBBAR,D.C.STRUPPA,A.VAJIAC,ANDM.B.VAJIAC
For real quadratic fields Q(√d), d > 0, the situation is very different. In this
case, the group of units is infinite and has the form
Od× = ±und n∈Z ≃Z/2Z×Z,
where ud >1 is the so-called fu(cid:8)ndam(cid:12)ental u(cid:9)nit. It is a difficult problem to find ud.
(cid:12)
For real quadratic fields K = Q(√d), d > 0, the logarithm of the fundamental
unit is called the regulator. For imaginary quadratic fields, the regulator is 1, as it
is for the ring of (rational) integers Z. For example, if d=2, u =1+√2 and the
2
regulator R of Q(√2) is
2
R =log(1+√2),
2
where 1+√2 is the fundamental solution of the Pell equation
x2 2y2 =1.
−
Let K be a quadratic field with discriminant ∆, so that ∆=d, if d 1 mod 4
≡
or 4d if d 2,3 mod 4. The quadratic character of K is the morphism
≡
∆
χ :(Z,+) C, χ (m):= .
K K
→ m
(cid:18) (cid:19)
It is a fundamental property that χ is periodic with period ∆.
K
| |
For K =Q(√d), we canalsodefine the Dirichlet character,alsodenotedby χ ,
K
by
χK : Z ∆Z × C×,
| | →
(cid:0) (cid:14) md (cid:1), if d≡1 mod 4
χK(m+|∆|Z)= ((cid:0)(−−11(cid:1)))mm22−8−11(cid:0)(cid:18)md((cid:1)md2,)(cid:19), iiff dd≡≡32 mmoodd 48
In the case K = Q(i),Q((−i)1)=(mZ−1[i)8(]m,+t5h)e(cid:18)r(imnd2)g(cid:19)o,f Gifauds≡sia6n imntoedge8rs, and the
O
Dirichlet character is
χQ(i) : Z 4Z × C×, χQ(i)(m)=( 1)m2−1 .
→ −
We can formulate th(cid:0)e (cid:14)deco(cid:1)mposition of rational primes as follows. Let p be a
rational prime. The decomposition of (p) in is given by:
K
O
pp,where N(p)=N(p)=p, if χ (p)=1
′ ′ K
p = (p),where N((p))=p2, if χ (p)= 1
OK K −
(p)2,where N((p))=p, if χ (p)=0
K
ThequadraticL-functionofaquadraticfieldK withdiscriminant∆isgivenby:
L(χK,s)= χK(n)n−s = 1 χK(p)p−s −1 .
−
n 1 pprime
X≥ Y (cid:0) (cid:1)
ZETA FUNCTION FOR MULTICOMPLEX ALGEBRA 7
The sum is defined for Re(s) > 1, but actually the quadratic L-function extends
analytically to Re(s) > 0. The value of L(χ ,1) is a remarkable one (see e.g. [6,
K
10, 11]).
Proposition 2.2. (i) For a real quadratic field K:
∆ 1
1 | |− πr
L(χ ,1)= χ (r)log sin .
K K
− ∆ ∆
| | Xr=1 (cid:18) (cid:18)| |(cid:19)(cid:19)
(ii) For an imaginary quadratipc field K:
∆ 1
π | |−
L(χ ,1)= χ (r)r.
K −|∆|23 Xr=1 K
3. The Dedekind zeta function
The Dedekind zeta function of a quadratic field K is given by:
ζK(s)= N(a)−s = 1 N(p)−s −1 . (3.1)
−
a p
X Y(cid:0) (cid:1)
Thesumis overallnon-zeroidealsaof andthe productisoverallprimeideals
K
O
of . If denote by
K
O
a =# a N(a)=n ,
n
{ }
then (cid:12)
(cid:12)
a
n
ζ (s)= .
K ns
n>0
X
The fundamentalproperty ofthe Dedekind zeta function is the following factor-
ization:
Theorem 3.1. For Re(s)>1, we have
ζ (s)=ζ(s)L(χ ,s),
K K
where s ζ(s) is the Riemann zeta function.
7→
We give below an idea of the proof. For Re(s)>1 we can write:
ζK(s)= 1 N(p)−s −1 , (3.2)
−
Yp pY|(p)(cid:0) (cid:1)
where (p) = p is the ideal generated by p in . Furthermore, for Re(s) > 1
K K
O O
we have:
ζ(s)L(χ ,s)= (1 p s) 1(1 χ(p)p s) 1
K − − − −
− −
p
Y
Lemma 3.2.
1 N(p)−s =(1 p−s)(1 χK(p)p−s).
− − −
pY|(p)(cid:0) (cid:1)
8 A.SEBBAR,D.C.STRUPPA,A.VAJIAC,ANDM.B.VAJIAC
Proof. Indeed, for a given rational prime p, if χ (p)=1, then
K
(p)=p K =pp′, N(p)=N(p′)=p,
O
and both sides are (1 p s)2. If χ (p) = 1, then p is inert and p = (p), with
− K
− −
N(p)=p2, and both sides are
(1 p2( s))=(1 p s)(1+p s).
− − −
− −
Finally, if χ (p)=0, then both sides are 1 p s. (cid:3)
K −
−
From this lemma, we obtain
ζK(s)= 1 N(p)−s −1
−
Yp pY|(p)(cid:0) (cid:1)
= (1 p s) 1(1 χ (p)p s) 1
− − K − −
− −
p
Y
=ζ(s)L(χ ,s).
K
A beautiful consequence of what we have seen is the Dirichlet class number
formula for imaginary quadratic fields. It says that:
2πh
=L(χ ,1),
K
w ∆
| |
whereh istheidealclassnumberofK,andw is thenumberofrootsofunityinK.
p
We turn now to the case K = Q(i), which is our main interest. The Dedekind
zeta function of Q(i) is given by:
ζQ(i) =ζ(s)L(χ 4,s) (3.3)
−
1 1 1
= ,
1 2 s (1 p s)2 1 p 2s
− − p≡1(Ymod4) − − p≡3(Ymod4) − −
where, in order to simplify the notations, we write χ 4 instead of χQ(i), since 4
is the discriminant of the field Q(i). The Dirichlet L-s−eries is then: −
χ (n)
4
L(χ−4,s)= −ns .
n 1
X≥
Note that χ is the character of Z, of period 4, given by:
4
−
0, if n 0 mod 4
≡
1, if n 1 mod 4
χ−4(n)= 0,1, iiff nn≡≡23 mmoodd 44 .
− ≡
This implies that
1 1
L(χ ,s)= .
−4 (4n+1)s − (4n+3)s
n 0 n 0
X≥ X≥
To study the analytic continuation of L(χ ,s) we relate it to the Hurwitz zeta
4
−
function, defined by:
1
ζ(s,α)= , Re(s)>1, α>0.
(n+α)s
n 0
X≥
ZETA FUNCTION FOR MULTICOMPLEX ALGEBRA 9
This function reduces to the Riemann zeta function for α=1. The general theory
says that s ζ(s,α) admits a meromorphic continuation to the whole plane, with
7→
a simple pole at s=1 with residue 1. We need the special value
1 Γ(α)
′
lim ζ(s,α) =
s 1 − s 1 −Γ(α)
→ (cid:18) − (cid:19)
where Γ is the Euler function (see e.g. [4]). An immediate consequence of this fact
is that s L(χ ,s) is analytically continuable to the whole plane, according to
4
7→ −
the equality:
1 3
L(χ 4,s)=4−s ζ s, ζ s, .
− 4 − 4
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
Another representation of L(χ ,s) is
4
−
∞ ( 1)n
L(χ ,s)= − =β(s),
−4 (2n+1)s
n=0
X
where β is the Dirichlet Beta function.
Thefunctions ζ(s)L(χ ,s)extendsmeromorphicallytoallC,withasimple
4
7→ −
pole at s=1. We then have, for 0< s 1 < :
| − | ∞
C
ζ(s)L(χ 4,s)= −1 +C0+C1(s 1)+C2(s 1)2+
− s 1 − − ···
−
We give below the values of C and C . From the expansions:
1 0
−
1
ζ(s)= +γ+γ (s 1)+
1
s 1 − ···
−
where γ is the (small) Euler constant, and
L(χ ,s)=L(χ ,1)+L(χ ,1)(s 1)+
4 4 ′ 4
− − − − ···
we obtain
L(χ ,1)
4
ζ(s)L(χ 4,s)= − +L′(χ 4,1)+γL(χ 4,1)+
− s 1 − − ···
−
Hence L(χ ,1) is the residue of ζ(s)L(χ ,s) at s=1 and
4 4
− −
γQ(i) :=L′(χ 4,1)+γL(χ 4,1)
− −
is what we may call the Euler constant γQ(i) of the field Q(i) (as γ =γQ).
1 3
L(χ ,1)= lim4 s ζ s, ζ s,
4 −
− s 1 4 − 4
→ (cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
1 1 1 3 1
= lim ζ s, lim ζ s,
4 s 1 4 − s 1 −s 1 4 − s 1
(cid:18) → (cid:18) (cid:18) (cid:19) − (cid:19) → (cid:18) (cid:18) (cid:19) − (cid:19)(cid:19)
= 1 Γ′ 43 Γ′ 41 .
4 Γ (cid:0)34(cid:1) − Γ (cid:0)14(cid:1)!
(cid:0) Γ(cid:1) (cid:0) (cid:1)
′
The logarithmic derivative of the Γ-function is a remarkable function. We only
Γ
need to know it satisfies the functional equation
Γ(z) Γ(1 z)
′ ′
− =πcot(πz).
Γ(z) − Γ(1 z)
−
10 A.SEBBAR,D.C.STRUPPA,A.VAJIAC,ANDM.B.VAJIAC
Hence
L(χ ,1)= 1 Γ′ 41 Γ′ 34 = π .
−4 −4 Γ (cid:0)41(cid:1) − Γ (cid:0)43(cid:1)! 4
To find the Euler constant γQ(i) of the(cid:0)fie(cid:1)ld Q(i(cid:0)), w(cid:1)e observe that:
γQ(i) L′(χ 4,1)
=γ+ − ,
L(χ ,1) L(χ ,1)
4 4
− −
which is what it is called the Sierpinski constant (see e.g. [8]):
L(χ ,1) Γ 3 2
γ+ ′ −4 =log 2πe2γ 4
L(χ 4,1) Γ(cid:0)1(cid:1)2!
− 4
= π log4+2(cid:0)γ (cid:1) 4 ∞ log 1 e−2πk
3 − − −
k=1
X (cid:0) (cid:1)
=0.8228252...
Hence the analogue of the Euler constant for Q(i) is:
γQ(i) =L′(χ 4,1)+γL(χ 4,1)
− −
π Γ 3 2
= log 2πe2γ 4
4 Γ(cid:0)1(cid:1)2!
4
π 3 1
= γ+log2+(cid:0) l(cid:1)ogπ 2logΓ ,
2 2 − 4
(cid:18) (cid:18) (cid:19)(cid:19)
where we used the classical complement formula for the Γ-function:
π
Γ(z)Γ(1 z)= , z =0, 1, 2,...
− sin(πz) 6 ± ±
Remark 3.3. We would like to give an idea on how to calculate the coefficients of
the Dirichlet series of ζQ(i)(s). First, for Re(s)>1, we have:
1 1 1 ∞ r2(n)
ζQ(i)(s)= 4 (m2+n2)s = 4 ns ,
(m,nX)6=(0,0) nX=1
where
r (n)= (p,q) Z Z, p2+q2 =n
2
{ ∈ × }
is the number of representatio(cid:12)n of n as a sum of two squares(cid:12).
(cid:12) (cid:12)
Secondly, we have:
∞ 1 ∞ χ 4(n)
ζQ(i)(s)=ζ(s)L(χ−4,s)= ns! −ns !
n=1 n=1
X X
χ (d)
4
−
= ∞ χ−4(n2) = ∞ Xd|n ,
nsns ns
n1X,n2=1 1 2 nX=1
