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.
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