INTRODUCTION

C"(0,A,s) = n£c"(0,A( U)

3- » ^ U

n

A « \

n

(9)

l^z,/c^n,27^fc

£

K"(0,/

(J* ) J ,*) " 2C"(0, A*'),*) -

2"(UfcU)

+

2n

£

l^i,k,p^n,i^k,k^p,p^i

C"(o,

^(/c)

,x)-3C"(o,^,:r)

-3C/,(0,A^,x)-3C,,(0,A^),x)

Here the second and the third terms have the meaning of "correction terms" and

can be written explicitly. In chapter I, we will also prove an asymptotic expansion

of the multiple gamma function.

For r linearly independent vectors v\, v2, ..., vr G R

n

, put

C(VUV2,..- ,Vr)= £ t ^ i

£1,^2,... ,£

r

0 ,

and call C(vi,V2, •.. ,vr) an r-dimensional open simplicial cone with basis v\, V2 ,

..., vr. Let 7i, cr2, ..., crn be all the isomorphisms of F into R. For x G F, we

put x ^ =

a;"7*.

We embed F into R

n

by

F 3 x (xW,xW... , x ^ ) G R n .

n ) \

Let R

+

be the group of all positive numbers and Ep be the group of all totally

positive units of F. A theorem of Shintani states that there exist finitely many

open simplicial cones Cj, j G J with basis in Dp such that

(10) R ; = U

£ 6 E

+e(U

j 6 J

C

i

).

Let r(j) be the dimension of Cj and put

Cj = C(VJI,VJ2,..- , V ( j ) ) ' Vj!,Vj2,... ,vjr(j) G O F -

For z e Cj, we define the coordinate £;(z) G R + by

Z = Xi(z)Vji + X2(z)Vj2 + • . . + Xr(j){z)vjr(jy

Put ^(z ) = (#1(2), #2(2)?... ,xr(j)(z)). For a fractional ideal a of F, we put

i^(CJ-,a) = { z G a n C

j

|0a?i(z),x

2

(^),... ,xr(j)(z) ^ 1, 'rr(z) G Q

r ( j )

}.

Now let f be an integral ideal of F and let Cf denote the ideal class group of

F defined modulo fooioo2 • • • oon. Let ho be the class number of F in the narrow

sense and let ai, ci2, •.., Qh0 be integral ideals which represent narrow ideal classes.

For c G Cf, take aM so that c and a^f belong to the same narrow ideal class and put

R(Cj,c) =

{zeR(Cj,(aIAf)-l)\(z)a^

= c in Cf}.