INTRODUCTION

3

ooi, 2 ^ i fS n are ramified in K. Then the conductor of K as a class field over F

is of the form foo2 • • • ocn with an integral ideal f of F. For a G Gal(if/F), let

CM=

£

^()"a

a 5(-t-~)==cr

be the partial zeta function attached to cr. Here (—^—) denotes the Artin symbol.

We regard F as a subfield of R and assume that ooi corresponds to the identical

embedding of F into R. Then the Stark conjecture states

CONJECTURE

S. Assume n ^ 2. Then there exists a unit e of K such that

exp(-2C'(0,a)) =

eCT

/or every a G Gal(if/F).

In chapter II, we will explain a method to derive Conjecture S from (5) in

detail.4. This conjecture was discovered independently by Shintani when F is a

real quadratic field. Furthermore he obtained an explicit formula in terms of the

multiple gamma function for the derivative of the partial zeta function at s = 0

for an arbitrary totally real field. Since his result is of central importance for our

investigation, let us describe it in some detail.

Let

UJ

—

(UJI,(JJ2,

• • •

5

^r), ui 0 for 1 ^ i ^ r and x 0. We define the r-ple

Riemann zeta function by

Cr(s,v,x)= ] T (x + tt)-s.

Q=raicji-|-7712^2-I \-mrujr

Here (mi,rri2,... , wir) extends over all r-tuples of non-negative integers. By a

method which goes back to Riemann, Barnes showed that Cr(s, UJ, x) can be contin-

ued meromorphically to the whole s-plane and holomorphic at s — 0; furthermore

we have an elementary formula for (r(0,uj,x):

Cr(0,(u;i,u;2,... ,ujr),y]ujkxk)

(6) "-

1

= ( - l ) r ^2 w;1

~UJO*

-'-&:.

/I,... ,Zr^0, ZH \-lr=r

-l z2-i

r.ir-iBh(xi)Bh(x2)

Bir(xr)

where Bn(x) denotes the n-th Bernoulli polynomial. We put

- logpr(uj)= lim \-—(r(s1uj,x)\ + l o g x L

x^+o{ds I

0

J

— Qr{s,UJ,X)

s=0

Tr(x,uj)

l0g 7 - T - .

pr{UJ)

The function Tr(x,uj) is the r-ple gamma function introduced by Barnes ([Bar2]).

If r = 1, we have

FI(X.UJ) 1 ^,xs ,,x 1

X 1

\ \ J = — = r ( - ) e x p ( - - - logu;).

4Strictly

speaking, a conjecture with 0 e G Z in place of 2 will be derived.