4
INTRODUCTION
Shintani introduced a further generalization of the r-ple Riemann zeta func-
tion. Let A (a,ij) be an n x r-matrix with a^ 0 for all i and j . For
lx (#i,X2, ,#r)5 %i = 0, 1 ^ i ^ r, x 7^ 0 and s E C, we define
oo n \ r
m i , . . . , m
T
. = 0 i = l l i = l
For 1 ^ i ^ n, let A^ denote the 2-th row of A. By definition, we have
3 = 1
We can show that £(s, ^4, #) can be continued meromorphically to the whole s-plane
and holomorphic at s = 0. Shintani obtained that ([Shi3])
(7)
C(0,A,x) = -fc(0,A«,x),
77. * '
i = l
(8)
¥sas,A,x)
s=0
= £ bg
•*- r I / ^-v—i &ij%ji X^ilt i&ir))
i=l
P r ( ( & i l ,
5
& i r ) )
+^E^)JI^i.r(-i)
Here in (8), the last term is "the correction term" and we refer the reader to chapter
I, §2.3 for the precise definition; we simply note that it is of elementary nature. A
remarkable feature of (8) is that
C(0,A,x) =
^C(Q,A(i),x)
(8')
i=l
l^i,k^n,i^k
+1 E \C(0,(j{lXx)-C(0,A^,x)-C(0,A^,x)
Here a prime denotes the derivative with respect to s. The second term of the right-
hand side represents the correction term. In chapter I, we will give proofs of these
results together with a generalization to the second derivative of £(s, A, x) at 5 = 0.
In analogy to (8'), we have5
5 T h e formula (8') (resp. (9)) is tautological whe n n = 2 (resp. n 2, 3) withou t th e explicit
expressions of th e correction terms . Bu t t o write th e formulas in these forms is suggestive.
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