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,

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http://dx.doi.org/10.1090/surv/106/01