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
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
S. Assume n ^ 2. Then there exists a unit e of K such that
/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.
• • •
^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.
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):
= ( - l ) r ^2 w;1
/I,... ,Zr^0, ZH \-lr=r
where Bn(x) denotes the n-th Bernoulli polynomial. We put
- logpr(uj)= lim \-—(r(s1uj,x)\ + l o g x L
l0g 7 - T - .
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
\ \ J = — = r ( - ) e x p ( - - - logu;).
speaking, a conjecture with 0 e G Z in place of 2 will be derived.