THE CYCLOTOMIC ZETA FUNCTION
1. Dirichlet Characters
Let N be a positive integer. A Dirichlet character modulo N is defined initially
as a homomorphism
χ : (Z/N Z)× −→ C× .
Any such character determines a least positive divisor M of N such that the character factors as
π
χo
M
χ = χo ◦ πM : (Z/N Z)× −→
(Z/M Z)× −→ C× .
The integer M is the conductor of χ, and the character χo is primitive. Note that
if n ∈ Z is not coprime to N but is coprime to M then χo (n + M Z) is defined and
nonzero. Perhaps confusingly at first, we also use the symbol χ to denote χo lifted
to a multiplicative function on the integers,
(
χo (n + M Z) if gcd(n, M ) = 1,
χ : Z −→ C, χ(n) =
0
if gcd(n, M ) > 1.
Thus (the lifted) χ(n) need not equal (the original) χ(n + N Z), and in particular
χ(n) need not equal 0 even when gcd(n, N ) > 1.
Especially, if N > 1 then the trivial character 1 modulo N has conductor M = 1,
and the trivial character 1o modulo 1 is identically 1 on (Z/1Z)× = {0}, and this
character lifts to the constant function 1(n) = 1 for all n ∈ Z, even though the
original character 1 modulo N is undefined on the coset N Z in Z/N Z.
Fix a rational prime p.
• Let pd k N and Np = N/pd and e = φ(pd ) = pd−1 (p − 1).
• Let f denote the order of p + Np Z in (Z/Np Z)× .
• Let g = φ(Np )/f = φ(N )/(ef ).
Thus altogether ef g = φ(N ). Note that Np = N and e = 1 for all primes p other
than the finitely many prime divisors of N . For k = 0, · · · , f −1, there exist Dirichlet
characters modulo Np that take p to ζfk , where ζf = e2πi/f , and the number of
Dirichlet characters modulo Np that take p to 1 is the order of (Z/Np Z)× /hp+Np Zi,
which is φ(Np )/f = g. Thus for k = 0, · · · , f − 1, the number of characters
modulo Np that take p to ζfk is g, independently of k. Any Dirichlet character
modulo N that is not defined modulo Np takes p to 0. That is, for k = 0, · · · , f − 1,
there exist g Dirichlet characters modulo N that take p to ζfk , and any Dirichlet
character modulo N that does not take p to ζfk for some k takes p to 0.
2. Cyclotomic Arithmetic
Again let N be a positive integer. Let ζN = e2πi/N , and consider the cyclotomic
number field
K = Q(ζN ).
1
2
THE CYCLOTOMIC ZETA FUNCTION
For any rational prime p, let e and f and g be as above. Then p factors in OK as
pOK = (p1 · · · pg )e ,
[OK /pi : Z/pZ] = f for each i.
Especially, the primes p such that p = 1 mod N decompose completely in OK ,
pOK = p1 · · · pφ(N ) and OK /pi ≈ Z/pZ for each i
if p = 1 mod N ,
while the other extreme case is
pOK = p and [OK /p : Z/pZ] = φ(N )
if p is a primitive root modulo N .
The rational primes that ramify are precisely the prime divisors of N , except that
2 does not ramify if 2 | N but 4 - N . In the exceptional case N = 2 mod 4, in fact
Q(ζN ) = Q(ζN/2 ), so if we avoid redundant cyclotomic extensions Q(ζN ) where
N = 2 mod 4 then the primes that ramify are exactly the prime divisors of N .
3. Cyclotomic Galois Theory, Briefly
The Galois group of K = Q(ζN ) over Z is
a
G = {ζN 7−→ ζN
: a + N Z ∈ (Z/N Z)× }.
The fact that the Galois group is a subgroup of G is clear. On the other hand,
showing that the Galois group is all of G is not entirely trivial, though it can be
made elementary. Recall the decomposition N = Np · pd where pd k N . We freely
make the identifications
G = (Z/N Z)× = (Z/Np Z)× × (Z/pd Z)× .
Fix a rational prime p. The inertia and decomposition subgroups of p in G are
Ip = {1} × (Z/pd Z)× ,
Dp = hp + Np Zi × (Z/pd Z)× .
Thus |Ip | = e and |Dp | = ef .
The inertia field KI,p and the decomposition field KD,p of p are the intermediate
fields of K/Q corresponding to the inertia and decomposition subgroups of G.
Thus Q ⊂ KD,p ⊂ KI,p ⊂ K. The decomposition field is so named because p
decomposes there as pOD = p1,D · · · pg,D (the pi,D are ideals), and for each i there
is no residue field growth, meaning that [OD /pi,D : Z/pZ] = 1, and visibly there
is no ramification. The inertia field is so named because each pi,D remains inert
in OI , which is to say that pi,D OI takes the form pi,I rather than decomposing
further, but here there is residue field growth, specifically [OI /pi,I : OD /pi,D ] = f ,
and again there is no ramification. Finally, each pi,I ramifies in OK as pi,I OK =
pei but with no further decomposition and with no further residue field growth,
[OK /pi : OI /pi,I ] = 1.
4. The Dedekind Zeta Function and its Euler Product
The ring of integers of K is
OK = Z[ζN ].
Define the norm of a nonzero ideal a of OK to be
Na = |OK /a|.
Thus we tacitly assert without proof that the quotient is finite. We further assert
without proof that the norm is strongly multiplicative, and that
Np = pf
where p | p and f is the inertial degree as above.
THE CYCLOTOMIC ZETA FUNCTION
3
Definition 4.1. The Nth cyclotomic Dedekind zeta function is
X
Y
ζK (s) =
Na−s =
(1 − Np−s )−1 , Re(s) > 1.
a
p
The sum is taken over nonzero ideals of OK , and the product is taken over maximal
ideals.
For any p, compute that
fY
−1
Y
Y
(1 − Np−s )−1 = (1 − p−f s )−g =
(1 − ζfk p−s )−g =
(1 − χ(p)p−s )−1 ,
χ
k=0
p|p
where the product is taken over all characters χ modulo N , each character understood to be the underlying primitive character extended to a multiplicative function
on Z. As discussed above, χ(p) = ζfk for g characters χ modulo N , independently
of k, these characters being defined modulo Np while the characters χ modulo N
that are not defined modulo Np take p to 0 and thus contribute a trivial factor of 1
to the last product in the previous display. Overall, then, we have
YY
YY
ζK (s) =
(1 − χ(p)p−s )−1 =
(1 − χ(p)p−s )−1 ,
p
χ
χ
p
which is to say that the N th cyclotomic Dedekind zeta function factors as the
product of all Dirichlet L-functions modulo N ,
Y
ζK (s) =
L(χ, s).
χ
This justifies the assertion in the definition that ζK (s) converges for Re(s) > 1.
One consequence of these calculations is as follows. It is known that the function
L(1, s) = ζ(s), which is initially defined only for Re(s) > 1, extends to a meromorphic function on {Re(s) > 0} whose only singularity is a simple pole at s = 1,
and it is known that L(χ, s) for χ 6= 1 is analytic on {Re(s) > 0}. Thus ζN (s)
is
on {Re(s) > 0} with its only possible pole at s = 1. Recall that
Q meromorphic
−s −1
(1
−
Np
)
= (1 − p−f s )−g , and estimate for any positive real s that
p|p
X
g X
∞
∞
∞
X
(1 − p−f s )−g =
≥
p−nf s
p−nf gs ≥
p−nφ(N )s = (1 − p−φ(N )s )−1 .
n=0
n=0
n=0
Multiplying over p gives
ζN (s) ≥ ζ(φ(N )s),
s > 0.
Thus ζN (s) must have a pole at s = 1: if it didn’t then it would converge for
all s > 0, but the displayed inequality shows that it diverges at s = 1/φ(N ). The
fact that ζN (s) has a pole at s = 1 is the crux of Dirichlet’s proof that there are
infinitely many primes in any credible arithmetic progression.
The Dedekind zeta function of K is the generating function for a representation
number function,
∞
X
ζK (s) =
|{a : Na = n}| n−s .
n=1
4
THE CYCLOTOMIC ZETA FUNCTION
On the other hand, let d = φ(N ). Then
"
∞
Y
X
X
L(χ, s) =
χ
n=1
#
χ1 (n1 ) · · · χd (nd ) n−s .
n1 ···nd =n
And so
|{a : Na = n}| =
X
χ1 (n1 ) · · · χd (nd ).
n1 ···nd =n
In particular, for any prime p,
|{p : Np = p}| =
X
χ(p).
χ
The previous display is nothing other than the second orthgonality relation,
(
X
φ(Np ) if p ≡ 1 (mod Np ),
χ(p) =
0
otherwise.
χ
That is, the second orthogonality relation is a special case of a cyclotomic counting
formula.