On the values at integers of the Dedekind zeta function of a real quadratic field
Authors: Kramer David 
Published: June 2010

In 1976 Shintani gave a decomposition of the Dedekind zeta function,

ζ
κ
(
s
)

\zeta \kappa (s)

, of a totally real number field into a finite sum of functions, each given by a Dirichlet series whose meromorphic continuation assumes rational values at negative integers. He obtained a formula for these values, thereby giving an expression for

ζ
κ
(
−
n
)
,

\zeta \kappa ( - n),

,

n
=
0
,

1
,

2
,
…

n = 0,\,1,\,2, \ldots

. Earlier, Zagier had studied the special case of

ζ
(
A
,

s
)

\zeta (A,\,s)

, the narrow ideal class zeta function for a real quadratic field. He decomposes

ζ
(
A
,

s
)

\zeta (A,\,s)

into

Σ
A

Z
Q

(
s
)

{\Sigma _A}{Z_Q}(s)

, where

Z
Q

(
s
)

{Z_Q}(s)

is given as a Dirichlet series associated to a binary quadratic form

Q
(
x
,

y
)
=
a

x
2

+
b
x
y
+
c

y
2

Q(x,\,y) = a{x^2} + bxy + c{y^2}

, and the summation is over a canonically given finite cycle of “reduced” quadratic forms associated to a narrow ideal class

A
A

. He then obtains a formula for

Z
Q

(
−
n
)

{Z_Q}( - n)

as a rational function in the coefficients of the form

Q
Q

. Since the denominator of

ζ
(
A
,

−
n
)

\zeta (A,\, - n)

is known not to depend on the class

A
A

, whereas the coefficients of reduced forms attain arbitrarily large values, it is natural to ask whether the rational function in Zagier’s formula might be replaced by a polynomial. In this paper such a result is obtained. For example, Zagier gives
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="15120 zeta left-parenthesis upper A comma negative 2 right-parenthesis equals sigma-summation Underscript upper A Endscripts StartFraction b Superscript 5 Baseline minus 10 a b cubed c plus 30 a squared b c squared Over a cubed EndFraction plus StartFraction b Superscript 5 Baseline minus 10 a b cubed c plus 30 a squared b c squared Over c cubed EndFraction minus 21 b left-parenthesis a plus c right-parenthesis">

15120
ζ
(
A
,
−
2
)
=

∑
A

b
5

−
10
a

b
3

c
+
30

a
2

b

c
2

a
3

+

b
5

−
10
a

b
3

c
+
30

a
2

b

c
2

c
3

−
21
b
(
a
+
c
)

 

15120\zeta (A, - 2) = \sum \limits _A {\frac {{{b^5} - 10a{b^3}c + 30{a^2}b{c^2}}}{{{a^3}}}} + \frac {{{b^5} - 10a{b^3}c + 30{a^2}b{c^2}}}{{{c^3}}} - 21b(a + c){\text { }}

\]
while our result is
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="15120 zeta left-parenthesis upper A comma negative 2 right-parenthesis equals one half left-parenthesis sigma-summation Underscript upper A Endscripts minus sigma-summation Underscript theta upper A Endscripts right-parenthesis left-parenthesis 60 a squared minus 117 a b plus 76 a c plus 38 b squared minus 117 b c plus 60 c squared right-parenthesis">

15120
ζ
(
A
,

−
2
)
=

1
2

(

∑
A

−

∑

θ
A

)

(
60

a
2

−
117
a
b
+
76
a
c
+
38

b
2

−
117
b
c
+
60

c
2

)

15120\zeta (A,\, - 2) = \frac {1} {2}\left ( {\sum \limits _A - \sum \limits _{\theta A} } \right )\,(60{a^2} - 117ab + 76ac + 38{b^2} - 117bc + 60{c^2})

\]
, where

θ
\theta

is the narrow ideal class consisting of principal ideals generated by elements of negative norm. Starting with a representation of

Z
Q

(
1
+
n
)

{Z_Q}(1 + n)

due to Shanks and Zagier for

n
=
1
,

2
,

3
,
…

n = 1,\,2,\,3, \ldots

as a certain transcendental function of the coefficients of

Q
Q

, we also obtain the result that

ζ
(
A
,

1
+
n
)

\zeta (A,\,1 + n)

is given as the same sum of reduced quadratic forms as in the formula for

ζ
(
A
,

−
n
)

\zeta (A,\, - n)

, times the appropriate “gamma factor.” This gives a new proof of the functional equation of

ζ
(
A
,

s
)

\zeta (A,\,s)

at integer values of

s
s

, and suggests the possibility that one might be able to prove the functional equation for all

s
s

by finding some relation between

Z
Q

(
s
)

{Z_Q}(s)

and

Z
Q

(
1
−
s
)

{Z_Q}(1 - s)

. So far we have not found such a relation.