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.