The special values at negative integers of Dirichlet series associated with polynomials of several variables
Authors: Eie Min King 
Published: June 2010

Let

P
(
X
)

P(X)

be a product of

k
k

linear forms in

r
r

variables

X
1

,
…
,

X
r

{X_1}, \ldots ,{X_r}

as given by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row upper P left-parenthesis upper X 1 comma ellipsis comma upper X Subscript r Baseline right-parenthesis equals product Underscript j equals 1 Overscript k Endscripts left-parenthesis a Subscript j Baseline 1 Baseline upper X 1 plus midline-horizontal-ellipsis plus a Subscript j r Baseline upper X Subscript r Baseline plus delta Subscript j Baseline right-parenthesis comma 2nd Row upper R e a Subscript j i Baseline greater-than 0 comma upper R e left-parenthesis delta Subscript j Baseline plus sigma-summation Underscript i equals 1 Overscript r Endscripts a Subscript j i Baseline right-parenthesis greater-than 0 period EndLayout">

P
(

X
1

,
…
,

X
r

)
=

∏

j
=
1

k

(

a

j
1

X
1

+
⋯
+

a

j
r

X
r

+

δ
j

)
,

Re
⁡

a

j
i

>
0
,

Re
⁡

(

δ
j

+

∑

i
=
1

r

a

j
i

)

>
0.

\begin {array}{*{20}{c}} {P({X_1}, \ldots ,{X_r}) = \prod \limits _{j = 1}^k {({a_{j1}}{X_1} + \cdots + {a_{jr}}{X_r} + {\delta _j}),} } \\ {\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \operatorname {Re} {a_{ji}} > 0,\;\operatorname {Re} \left ( {{\delta _j} + \sum \limits _{i = 1}^r {{a_{ji}}} } \right ) > 0.} \\ \end {array}

\]
Suppose that

β
=
(

β
1

,
…
,

β
r

)

\beta = ({\beta _1}, \ldots ,{\beta _r})

is an

r
r

-tuple of nonnegative integers. Consider the zeta function
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z left-parenthesis upper P comma beta right-parenthesis left-parenthesis s right-parenthesis equals sigma-summation Underscript n Subscript 1 equals 1 Baseline Overscript normal infinity Endscripts midline-horizontal-ellipsis sigma-summation Underscript n Subscript r equals 1 Baseline Overscript normal infinity Endscripts n 1 Superscript beta 1 Baseline midline-horizontal-ellipsis n Subscript r Superscript beta Super Subscript r Superscript Baseline upper P left-parenthesis n 1 comma ellipsis comma n Subscript r Baseline right-parenthesis Superscript negative s Baseline comma upper R e s greater-than StartFraction r plus StartAbsoluteValue beta EndAbsoluteValue Over k EndFraction comma">

Z
(
P
,
β
)
(
s
)
=

∑

n

1
=
1

∞

⋯

∑

n

r
=
1

∞

n
1

β
1

⋯

n
r

β
r

P

(

n
1

,
…
,

n
r

)

−
s

,

Re
⁡
s
>

r
+

|

β

|

k

,

Z(P,\beta )(s) = \sum \limits _{{n_{1 = 1}}}^\infty { \cdots \sum \limits _{{n_{r = 1}}}^\infty {n_1^{{\beta _1}} \cdots n_r^{{\beta _r}}P{{({n_1}, \ldots ,{n_r})}^{ - s}},\qquad \operatorname {Re} s > \frac {{r + |\beta |}} {k}} } ,

\]
where

|

β

|

=

β
1

+
⋯
+

β
r

|\beta | = {\beta _1} + \cdots + {\beta _r}

.

Z
(
P
,
β
)
(
s
)

Z(P,\beta )(s)

has an analytic continuation in the whole complex plane and it is regular at

s
=
0
,

−
1
,

−
2
,
…
,
−
m
,
…

s = 0,\; - 1,\; - 2, \ldots , - m, \ldots

. In this paper, we shall compute the explicit values of

Z
(
P
,
β
)
(
s
)

Z(P,\beta )(s)

at

s
=
0
,

−
1
,

−
2
,
…
,

−
m
,
…

s = 0,\; - 1,\; - 2, \ldots ,\; - m, \ldots

and express them in terms of finite sums of polynomials in Bernoulli numbers.