CHAPTER I
MULTIPLE GAMMA FUNCTION
AND ITS GENERALIZATIONS
In §1, we will derive contour integral representations for the multiple Riemann
zeta function and for the multiple gamma function. The results are due to E.
W. Barnes [Barl], [Bar2]. We also define a generalization of the multiple gamma
function by considering a higher derivative of the multiple Riemann zeta function
at s = 0. In §2, we consider a generalization of the multiple Riemann zeta function.
We will derive the formulas of the value and the derivative at s = 0 for this zeta
function. The results are due to T. Shintani [Shil], [Shi2], [Shi3]. In §3, we will
derive the formula of the second derivative at s = 0 for the zeta function studied in
§2. In §4, we will study the asymptotic expansion of the multiple gamma function
due to Barnes. We will give an estimate for the remainder term of the asymptotic
expansion under a certain condition.
1. Basic integral representations
1.1. Let r be a natural number and let u (CJI, u;2, *v) R+5 x 0. We
define the r-ple Riemann zeta function by
(1.1) Cr(«,W,x)= Yl (* + fi)-.
£l=m\LO\ -\-m20J2 H \-mruJr
Here (mi,7712,... ,m
r
) extends over all r-tuples of non-negative integers. Let us
show that the series in (1.1) converges locally uniformly when SR(s) r. Put
J = 3?(s). Since
I £ (x +
siys\s
£ (x + ny,
Q=miui-\-m2^2-\ \-mrujr Q.=m\uj\+m2W2-\ \-mrojr
it suffices to prove that the series on the right-hand side converges locally uniformly
when s a r. First assume r = 1. Then, for a 1, we have
0 0 0 0 /.00
^jT {x + miLOi)'"7 = x~G + ^jT (x + ,rn1LJ1)~(J ^ x~a + (x + uit)~adt.
mi=0 mi = l ^ °
We obtain
0 (1.2 ) ^0(
l + 1

M
)-"s{l
+
_^-1J}I-.« „i.
m i = 0
Put
_ , ^ f l 1 1
a r,
13
http://dx.doi.org/10.1090/surv/106/01
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