数理解析研究所講究録
1091 巻 1999 年 256-297

256

ここに掲載された論考は、 1997 年 10 月 1 日逝去された故三井孝美先生の御遺稿です。 同

じ年の暮、整然と整理された研究室の書棚に、幾度も推敲を重ねたと思われる何編もの下書
きと共に、 ほぼ完成された原稿として遺されていました。 内容は ‘ Tokyo J. Math. に発表されたいくつかの論文の続編であり、病気快癒の後に投稿されるお積もりであったと推察されます。

三井先生は、常に解析数論研究の中心に位置され、 この分野の研究者から厚い人望と尊敬
を集め、

$1,|$

一ダー的存在として研究の発展、後進の育成 1r 尽くされまし$\neq-\text{。}$

先生が-.

そあ

研究生活を始めた当時は、 日本の解析数論の黎明記で研究者の数も少なく、 1969 年に数理解析研究所で開かれたこの分野で最初の研究集会は、雑誌 “数学” 誌上に研究代表者竜沢周雄教授によって 『小グループの集会』 と紹介される程でした。 この集会の講演者の–人であった先生は、以後、 日本の解析数論の第–人者として多くの研究集会に関わりました。 現

在、解析数論はその研究者を増やし活発に研究されるに至りましたが、その発展に先生が果たした役割の大きいことは、たとえば、数理解析研究所での三度の枡究集会で研究代表者を務められたことからも窺われます。
今回、 山梨大学の中井喜信教授のご尽力により、本稿が同研究所講究録に掲載され先生のご研究が公となり、原稿発見者として、また数論研究者の–人として深い喜びと安堵を感じます。

1999 年 1 月

学習院大学理学部

中野伸

註:\S 3 以下, \S の No., Theorem の No., 式 No. の . ) に, -部門れている所がありますが,そのままにしてありま凱 あるいは, そこを修正される御積りであったのかも知れまん。
$1*$

$*$

中井喜信

257

On Certain Multiple Series with Functional Equation
in aTotally Imaginary Number Field II

TAKAYOSHI MITSUI
Gakushuin University\S 1. Introduction
In the former paper [3] we treated the series of the type

$0 \neq(\mu)\mathfrak{a}0\sum_{\mathrm{c}\neq}\sum_{\nu\in \mathrm{b}}\frac{1}{N(\mu)^{1/2}}exp\{-2\pi S(|\mu \mathcal{U}|\mathcal{T})+2\pi iS(\mu \mathcal{U}\xi)\}$.

In this paper, we shall consider a multiple series of more general type (see (1.6) below)

and prove that it satisfies a transformation formula.
Before stating our main result, however, we shall have to explain some definitions andnotations.

Let$\phi_{a}$be mappings from$N^{a}$

to$N$$(a=1, \ldots , k)$, where$N$

is the set of positive rational

integers, and $\phi_{0}=2l$ an even number. Let $J$ be the set of $(k+1)$ -tuplesof positive integers satisfying the conditions$1\leq j_{1}\leq\phi_{0}=2l$$1\leq j_{2}\leq\phi_{1}(j_{1})$

$\{$

,

,

$1\leq j_{3}\leq\phi_{2}(j_{1},j_{2})$,$1\leq j_{k+1}\leq\phi_{k}(j1, \ldots,jk)$

In particular, we denote by$I$

the element of $J$ such that$j_{1}=j_{2}=\cdots=j_{k+1}=1$.

.

$J=(j_{1}, .

.\cdot.\cdot,j_{k+1})$

258

TAKAYOSHI MITSUIFor each $J\in J$ we put,$P_{J}=l \prod_{=a1}^{k}\phi_{a}(j1, \ldots,ja)$ス$\backslash \mathit{1}\mathrm{t}\perp.1)$(1.2)$(v^{7\backslash \mathrm{D}})=_{A_{j/’\mathit{1}’/J}}\backslash ^{A}\gamma t\supset r)\backslash$」,$(q=0, \ldots, k)$$\alpha_{J}^{(q)}=\sum_{a=0}\frac{j_{a+.1}.-1}{\phi_{a}(j_{1},\ldots,j_{a})\cdot\emptyset q(j1,\ldots,jq)}q$

and
(1.3)

.

$(q=0, \ldots, k)$$\alpha_{J}^{*(q)}=\sum_{0a=}^{q}\frac{\phi_{a}(j_{1},)j_{a})-j_{a+}1}{\phi a(j1,\ldots,j_{a})\cdot\emptyset q(j1,\ldots,jq)}\ldots.$.

For the case $q=k$ , we write$(k)$$\alpha_{J}=\alpha_{J}$$*(k)$,$\alpha_{J}^{*}=\alpha_{J}$.

We note that$\alpha_{I}=0$,$\alpha_{J},$$\alpha_{J}^{*}\geq 0$.

Let If be a totally imaginary number field of degree $n=2r,$$1,$$\ldots,$

by

$r)$

$\mu^{(q)}$the pairs of the complex conjugates of

the conjugates of$(\mu^{(1)}, \ldots, \mu^{(})n)$such that

$\mu$

$\mu$

is a number of If, then we denote$n$. More generally, we shall use -dimensional complex vector$n$$\xi_{p+r}=\overline{\xi}_{p}(p=1, \ldots, r)$

$U$

. If

in If $(q)(q=1, \ldots, n)$ . We define -dimensional vector $\mu=$be the unit group of$I\acute{\mathrm{t}}$k-th powers of the elements of$U$each other with respect toif$U^{k}$$\xi=(\xi 1, \ldots, \xi_{n})$and write$S( \xi)=\sum_{=q1}^{n}\xi_{q}$Let$I\zeta$$I\zeta^{(p)},$ $K^{(r+_{\mathrm{P}})}=\overline{I\acute{\mathrm{t}}^{(p)}}(p=$,$N( \xi)=\prod_{q=1}^{n}\xi_{q}$. We denote by$U^{k}$

the subgroup of. The non-zero numbers$\alpha/\beta$

is a number of.

$U^{k}$$\alpha,$$\beta$

. Let

$U$

consisting of theare said to be associated to$V$

be the differente ideal of

259

MULTIPLE SERIES IN A TOTALLY IMAGINARY NUMBER FIELD II$K,$

the absolute value of the discriminant of$N(\mathfrak{D})=D$

non-zero ideal
To eachwe put$a$$J$$a^{*}=(\emptyset 0)^{-1}$

, and

$R$

the regulator of $K$ . For.

in $J$ we associate non-zero fractional ideal

numbers such that $\tau_{p+r}=\tau_{p}(p=1, \ldots, r),$$\overline{\xi}_{p}(p=1, \ldots, \Gamma)$$I\acute{\mathrm{t}}$$\xi_{1},$$\xi_{n}$$\ldots,$$a_{J}$. Let$\tau_{1},$$\ldots,$$\tau_{n}$be positive

complex numbers such that $\xi_{p+r}.=$.

We shall often use the multiple summations and products written as follows:$\sum_{j_{1}=12}^{\emptyset}\sum^{\phi}0j=11(j1)\ldots\phi q(j_{q+}j_{1\sum^{)}},\ldots,j_{q}\iota=1’.

\prod_{j_{1}=1j_{2},\backslash }^{\emptyset}.\prod_{=1}^{(}\ldots\prod_{j_{q}}^{1j_{q}}0\emptyset 1(j1)\phi_{q}j_{1}+1’..=.1)$$(q=0, \ldots, k)$

For the sake of simplicity, we shall denote them by$\sum_{J}^{(q)}$,$\prod_{J}^{(q)}$,$\prod_{J}=\Pi^{(k}J)(=\prod_{j\in J})$,

respectively. For the case $q=k$ , we write$\sum_{J}=\sum_{J}^{(k)}(=\sum_{J\in J})$.

Using the notations above, we put(1.4)$M_{a}= \prod_{J}^{(a-}(\phi 1)a)1/(\phi 0\cdots\phi_{a-1})$$(a=1, \ldots, k)$

and

(1.5)

$P=\iota M_{1}\cdots Mk$.

Now we define the series as follows :(1.6)$M(\tau, \xi;\alpha_{J}, aJ)=M(\tau_{1}, \ldots \mathcal{T}r’\xi_{1})’\ldots,$ $\xi r;\{\alpha_{J}\}J’\{a_{J}\}_{J})$$= \sum\cdots\sum$$\sum$$\prod N(\nu_{J})^{-\alpha_{j}}$$0\neq\nu_{J}\in \mathrm{Q}J0\neq\nu.\in t\emptyset IJ$$\nu_{J}/U^{\phi(J)}$,$(J\neq I)$$\cross\exp\{-2P\pi s(\mathcal{T}\prod_{J}|\nu J|1/P_{J})+2P\pi is(\xi\prod_{J}\nu_{J}1/P_{J)}\}$

,

.

260

TAKAYOSHI MITSUI

where the multiple sum is taken overnumbers in$a_{J}$$\{\nu_{J}\}_{J\in J}$; for each $J(\neq I),$not associated to each other with respect tonon-zero numbers in$a_{I}$$U^{\phi(J)}$$\nu_{J}$runs through non-zero, and$\nu_{I}$runs through all.

We shall show that the series (1.6) is well-defined We replace each $\nu_{J}(J\neq I)$ by$\nu_{J}\epsilon^{\gamma}\wedge(7J^{\vee}\backslash ,$($.\epsilon_{J}\in$

しりし.

$\overline{-1^{\mathrm{I}}1}1\mathrm{e}\mathrm{n}$$N(\nu_{J}\epsilon^{\phi(J)})J(=N\nu_{J})$and$\prod_{J}(_{\mathcal{U}_{J}}\epsilon_{J}^{\psi(})J)1/P_{J}=\nu_{I}^{1/P}\prod_{\neq I}x(\nu^{1}\epsilon)JJ/P\phi J(JJ)/PJ$.

By (1.1),$\frac{\emptyset(J)}{P_{J}}=\frac{1}{P_{I}}\frac{P_{I}}{(P_{I},P_{J})}$,

SO$\prod_{J}(\nu_{J}\epsilon_{J}^{\emptyset(j}))1/P=J\mathrm{I}\mathrm{I}^{\nu^{\prime P}(\nu}1JJI\mathrm{I}\mathrm{I}\epsilon^{P/(PP})II’ J)J1/P_{t}$.

This shows that$\nu_{I}\prod_{IJ\neq}\epsilon JP_{I}/(P_{I},P_{J})$runs through all non-zero elements of$a_{I}$

with$\nu_{I}$. Hence the sum over$\nu_{I}$changed.
LetLet$\epsilon_{1},$$\ldots,$$\epsilon_{r-1}$$\omega=e^{2\pi i/L}$

be the fundamental units of, where$L=1\mathrm{c}\mathrm{m}_{J\in \mathcal{J}}\{P\}j$$I\acute{\mathrm{t}}$We denote $\rho=e^{2\pi i/w}$ .

.

We define the sum
L-l$T(\tau,$$\xi_{1}\alpha J’ a_{J})=\sum\cdots$$h_{1}=0$L-l$\sum M(\tau,$ $\xi_{1}\omega^{h}1,$ $\ldots,$$\xi_{r}\omega^{h_{\mathrm{r}}}|\alpha_{J},$$h_{f}=0$

We put(1.7)$N_{a}= \prod_{J}^{)}(\emptyset(a-1a)-1+2\alpha_{J}^{(a}-\cdot 1)$$(a=1, \ldots, k)$$a_{J})$.

remains un-

261

MULTIPLE SERIES IN A TOTALLY IMAGINARY NUMBER FIELD II

and
(1.8)

$(a=1, \ldots, k)$$N_{a}^{*}= \prod_{J}^{(a-}(\phi a)-1+2\alpha_{J}^{*}-)1)(a1$For$\tau_{1},$$\ldots,$$\tau_{n}$and$\xi_{1},$

we define$\xi_{n}$$\ldots,$$(p=1, \ldots, r)$$\tau_{p}^{*}=\frac{\tau_{p}}{\tau_{p}^{2}+|\xi_{p}|^{2}}$

(19)

,

,

$\{$

$\xi_{p}^{*}=\frac{\xi_{p+r}}{\tau_{p^{2}}+|\xi p|^{2}}$

Let

$w$

,

$\xi_{p+r}^{*}=\frac{\xi_{p}}{\tau_{p}^{2}+|\xi_{p}|^{2}}$$(p=1, . \ldots, \Gamma)$.

be the number of the roots of unity in $K$ . We put$d_{I}=(w, P_{I})$,$d_{J}=(w, \phi(J))$$(J\neq I;J\in J)$.

We put(1.10)$s_{J}=(1-\alpha_{J})PJ$$s_{J}^{*}=(1-\alpha_{J}^{*})P_{J}$’$(J\in J)$.

We introduce the series as follows:$\zeta(s, a)=\sum_{0\neq(\mu)\subset\emptyset}\frac{1}{N(\mu)^{s}}$$(_{S=\sigma+}it, \sigma>1)$,

where the sum is taken over all non-zero principal ideals contained in . This function$\mathfrak{a}$$((s, a)$ of$s$

is continued analytically on the whole -plane. (See [3] or \S 2 below)$s$

We now state Main Theorem:
MAIN THEOREM.

$T(\tau, \xi;\alpha a_{J}J’)$$\frac{\prod_{J}N(a_{J})^{1/2}}{(N_{1}\cdots N_{k})^{r}/2}\prod_{p=1}^{r}(\tau_{pp}+\}\xi|2)^{1/4}2$.

sat isfies a transforma tion formula as follows:$\{T(\tau, \xi;\alpha_{j}, \alpha J)-S(\mathcal{T}, \xi;\alpha_{J}, a_{j})\}$$= \frac{\prod_{J}N(a_{J}^{*})1/2}{(N_{1}^{*}\cdots N_{k}^{*})^{r/}2}\prod_{p=1}r(T_{p}^{*}+2|\xi \mathrm{P}*|^{2})^{/4}1$.

$\{T(\tau^{*},\xi*;\alpha_{J’ J}\emptyset)*-S(\tau\xi*,*J;\alpha^{*}, a_{j})*\}*$

,

262

TAKAYOSHI MITSUI

where(1.11)$S(\tau, \xi).)\alpha_{J},a_{J}$$= \frac{(2^{2}\pi)^{r}}{d_{I}\sqrt{D}}\{p_{I}\prod_{J}\phi(J)\}^{r-}1Lr\prod d_{J}J$;$\frac{|\dot{\zeta}_{p}|\wedge}{\tau_{p}^{2}+|\xi_{p}|^{2}}l\cdot\cdot\backslash$$\cross\sum_{J}\{’\frac{\overline{P}_{J}\mathit{1}-(\prime\backslash \wedge\hat{\Delta}S_{J})}{N(a_{J})(2^{2}P\pi)^{ns_{J}}}.‘\prod p=1f\frac{\mathrm{i}}{(_{\mathcal{T}_{p}^{2}+}|\xi_{p}|^{2})^{S_{J}}}F(S_{J’J’)}\frac{\mathrm{i}}{2}-S1$$aj_{1})\}$$\cross\prod_{J(\iota\neq)}J\iota\in Jj\zeta(1+\frac{s_{J}-s_{J_{1}}}{P_{J_{1}}},$

and$S(\tau^{*})\xi**a^{*};\alpha_{J},J)$$= \frac{(2^{2}\pi)^{r}}{d_{I}\sqrt{D}}\{P_{I}\prod_{J}\emptyset(J)\}^{r}-1Lr\prod d_{J}J$$\cross\sum_{J}\{\frac{P_{J}\Gamma(2_{S_{J}^{*}})r}{N(\alpha_{J}^{*})(2^{2}P\pi)^{n}s_{J}*}\prod_{p=1}^{r}\frac{1}{(\tau_{p^{*^{2}}}+|\xi_{p}^{*}|^{2})^{s_{j}}*}F(s_{J}^{*},$$\cross\prod_{J_{1}\in J}\zeta(1+\frac{s_{J}^{*}-s_{J_{1}}^{*}}{P_{J_{1}}},$$\frac{1}{2}-SJ’ 1*$$a^{*}J_{1)}\}$;$\frac{|\xi_{p}^{*}|^{2}}{\tau_{\mathrm{P}}^{*2}+|\xi p|^{2}*})$,$(J_{1}\neq J)$$F(\alpha, \beta,\gamma;x)$

being the Gauss hypergeometric function.

First we shall consider, in \S 2, the zeta functions$\zeta(s, \lambda;\emptyset)$and summarize some prop-

erties of them in Theorems 2.1 and 22.
In \S 3, we shall consider the function $G(s, x;\iota)$ that was first studied by Rademacher[4], and prove two theorems. We shall have to estimate $G(s, x;\iota)$ in wider range (Theorem

32).

Next in \S 4, by applying the transformation formula of Hecke-Rademacher, we shallobtain the representation of$T(\tau, \xi;\alpha_{J}, a_{J})$as the series of the complex integrals:$T( \tau, \xi;\alpha J’ aJ)=A\sum\frac{1}{2\pi i}I(\sigma 0))H\lambda(s,\mathcal{T},$$\xi;\alpha J’\alpha_{J}dS\lambda$

263

MULTIPLE SERIES IN A TOTALLY IMAGINARY NUMBER FIELD II(see (4.17) and (4.18)).

Using theorems in \S 2 and \S 3, we shall have the estimate offunctional equation satisfied by$H_{\lambda}$($s,$ $\tau,$ $\xi;\alpha_{J}$, a ) and the$J$(Theorems 5.1 and 5.2).

$H_{\lambda}(s, \tau, \xi;\alpha J’ a_{J})$

These reslults and the residues will give, in \S 6, the proof of Main Theorem.

\S 2. Zeta functions with Gr\"ossencharacters
We introduce the zeta function with Gr\"ossencharacter.

:$\lambda$$(s=\sigma+it, \sigma>1)$$\zeta(s, \lambda;a)=\sum_{0\neq(\mu)\subset a}^{\cdot}\frac{\lambda(\mu)}{N(\mu)^{s}}$,where the sum is taken over all non-zero principal ideals contained in ,$\alpha$$\lambda(\mu)=.p1\prod_{=}^{r}|\mu^{(p}|^{-})iV\prod_{=}^{n}\mathrm{p}q1(\frac{\mu^{(q)}}{|\mu^{(q)}|})^{a_{q}}$

and
(2.1)
$\{e_{p}^{(j)}\}$$v_{p}= \sum_{j=1}^{r-}1e_{p}^{(j})(2\pi m,$.

are the numbers satisfying the following equations:$(j=1, \ldots, r-1)$ ,$\sum_{p=1}^{r}e_{p}^{(j)}=0$

1$i.=.j$,$0$$i\neq j$,$\sum_{\mathrm{p}=1}^{r}e\log|\mathcal{E}_{j}|=p(i)(_{\mathrm{P})}\{$

where$(p=1, \ldots, r)$$+ \sum_{q=1}^{n}a\arg\epsilon_{j)}^{(q})q$$m_{1},$$\ldots$,$m_{r-1}$$(i,j=1, \ldots, r-1)$ .

are the rational integers and$a_{1},$$\ldots,$

tegers such that $a_{p}\cdot a_{p+r}=0(p=1, \ldots, r)$ . Further
condition$\prod_{q=1}^{n}\rho^{(}q)^{a}q=1$(See [3, p.63].)

Here we note that$\sum_{p=1}^{r}v_{p}=0$.

.

$a_{n}$$a_{1},$

are non-negative rational in$\ldots,$$a_{n}$

satisfy the additional

264

TAKAYOSHI MITSUITHEOREM 2.1.

(1)$((s, \lambda;\alpha)h$as the analytic continuati on over the$\iota vh_{\mathit{0}}l\mathrm{e}$s-planeand satisfies the functional equati on as follows;(22)$\zeta(s, \lambda;a)=\frac{(2\pi)^{n(s-}1/2)}{N(a)\sqrt{D}}\frac{\Gamma(1-S,\overline{\lambda})}{\Gamma(s;\lambda)}.\zeta(1-S,\overline{\lambda}).a)*$,$1X\Gamma\underline{h}\rho r\rho$$\Gamma(s ; \lambda)=\prod_{p=1}r\tau(S+\frac{iv_{p}}{2}+\frac{a_{p}+a_{p+\Gamma}}{2})$

(2)

If

$\lambda\neq 1$.

, then$\tau(s;\lambda)\zeta(_{S,\lambda;a)}$$is$an entire function.
(3)

In the case$\lambda=1,$$\zeta(s, 1;a)=\zeta(s, a)$and$\Gamma(s)r\zeta(S, a)$is a$\mathrm{m}$eromorphic function with only two simple poles at $s=0$ and 1.

(4)$\zeta(s, a)$is regular in the whole -plane except at $s=1$ , where $((s, a)$ has a simple$s$pole with the residue$\frac{(2\pi)^{r}R}{wN(a)^{\sqrt{D}}}$

PROOF.

(See [3, Lemma 2.1].)

THEOREM 2.2.

口In the strip$-1/2 \leq\sigma\leq 1+\max\{P_{J}\}$,$J\in J$

we have(2.3)$\zeta(_{S\lambda;},a)(_{S}-1)\mathrm{e}(\lambda)\ll(1+|t|)^{2}n$where$e(\lambda)=\{$

1 if
if

$0$

and the constants inpli$e\mathrm{d}$in this est$\mathrm{i}m$$\lambda=1$,,$\lambda\neq 1$

ation (2.3) depend on$\lambda$

and a.

265

MULTIPLE SERIES IN A TOTALLY IMAGINARY NUMBER FIELD IIPROOF.

$\mathrm{I}\mathrm{f}-1/2\leq\sigma\leq 3/2$

$1+ \max_{J\in J}\{P_{J}\}$, then (2.3) is true ([4, Hilfssatz 15]). If $3/2\leq\sigma\leq$, then $\zeta(s, \lambda;a)=o(1)$ . Thus Theorem is proved口\S 3. Hypergeometric functions
Let

$l$

$0\leq x<1$

be a non-negative number,$s$

a complex number anda real number such that

. We put(3. 1)$F(s, l ; x)=F( \frac{s+.l}{2},$ $\frac{1-s+l}{2},$ $l+1$

where

$x$

$F(\alpha, \beta,\gamma;x)$; $x)$ ,is the Gauss hypergeometric function, and(32)$G(s, l ; x)= \frac{\Gamma(s+^{\iota)}}{2^{s+\tau(\iota}\iota+1)}F(s, l ; x)$.

It is known that $F(s, \iota;x)$ is an entire function and that $G(S, \iota_{;x})$ is regular in thehalf-plane $\sigma>0$ ([4, p.368]). From (3.1) and (3.2) we easily see that
(3.3)

$F(1-s, l ; x)=F(s, l ; x)$ ,(3.4)$G(1-s, \iota;x)=\frac{\Gamma(1-S+l)}{\Gamma(s+^{\iota)}}2^{2s-1}G(s, l;X)$,

which shows that $G(s, l;x)$ is meromorphic in the whole s-plane.

THEOREM 4.1.

Let $m$ be non-nega$- \frac{l+1}{2}$$ti\iota\prime e$ra tional integer. If\dagger $m< \sigma<\frac{l+3}{2}+m$ ,then $G(s, l;X)$ is represented by an integral as follows:(3.5)$G(s, l;x)= \frac{1}{2\sqrt{\pi}}\frac{\Gamma(\frac{s+l}{2})}{\Gamma(\frac{l-s\dagger 3}{2}+m)}\int_{0}^{\infty}\frac{v^{(+S-})/2-mg_{m}(\iota 1v)}{(1+v)(\iota-s)/2+2(1+v-X)(\iota+S)/2+m+1}dv$

,

266

TAKAYOSHI MITSUI

where $g_{m}(v)$ is apolynomial of degree $\leq m+1$ and $g_{m}(0)\neq 0$ .

PROOF.

(3.6)

By [4, (3.322)], we obtain$G(s, l;x)= \frac{\underline{\rceil}}{2\sqrt{\pi}}\frac{\Gamma(\frac{s+l}{\underline{9}}1\backslash \text{ノ}{\Gamma(\frac{l-s+3}{2})}}J_{0}^{\gamma}\infty\frac{\eta)(\iota+s-1)/2(\mathit{0}\cap’)d\backslash ^{\tau}\text{ノ}{(1+v)(l-S)/2+2(1+v-X)(l+s\rangle/2+1}}.dv$

,

where
$g_{0}(v)=(l+1)(1+v-x)-(l+s)xv/2$ .

(3.7)

The integral in the right-hand side of (3.6) is convergent for$-(l+1)/2<\sigma<(\iota+3)/2$

and we see that$g_{0}(0)=(l+1)(1-x)\neq 0$ ,which shows that Theorem is true for $m=0$ .
Assume that Theorem is true for $m(>0)$ and that$-(l-1)/2+m<\sigma<(\iota+3)/2+m$ .

By the integration by parts, the integal of the right-hand side of (3.5) is$I= \int_{0}^{\infty}\frac{v^{(1)/}-2-mg_{m}(\iota+Sv)}{(1+v)^{2}+(l-S)/2(1+v-X)(\downarrow+S)/2+m+1}dv$$= \int_{0}^{\infty}(\frac{1}{1+v})^{(-\theta+}\iota 5)/2+m\frac{v^{(+)}-1/2-m(ls1+v)m+1/2g_{m}(v)}{(1+v-X)(\iota+\theta)/2+m+1}dv$$=[ \frac{-1}{(l-S+3)/2+m}(\frac{1}{1+v})^{(\iota \mathrm{a}}-s+)/2+m\frac{v^{()}-1/2-m(l+s1+v)m+1/2g_{m}(v)}{(1+v-X)^{(}l+s)/2+m+1}]_{0}^{\infty}$$+ \frac{1}{l-s+3}\int_{0}^{\infty}\frac{1}{(1+v)(l-s+3)/2+m}\frac{d}{dv}\{\frac{v^{(\mathrm{t}+s-1)/2m}-(1+v)m+1/2g_{m}(v)}{(1+v-x)(\iota+\theta)/2+m+1}\}dv$$\overline{2}+m$

267

MULTIPLE SERIES IN A TOTALLY IMAGINARY NUMBER FIELD II$= \frac{1}{\frac{l-s+3}{2}+m}\int_{0}^{\infty}\frac{1}{(1+v)^{()/}l-S+32+m}\frac{d}{dv}\{\frac{v^{()}l+s-1/2-m(1+v)m+1/2gm(v)}{(1+v-X)(\iota+S)/2+m+1}\}dv$.

The last integrand is$\frac{v^{(+S-}\iota 3)/2-mg_{m+1}(v)}{(1+v)(\iota-s)/2+2(1+v-X)(\iota+S)/2+m+2}$

,

where
(3.8)

$g_{m+1}(v)=\{-(m+1)v^{2}+((l+s-3)/2-2m-x(l+s)/2)v$$+((l+s-.1)/2-m)(1-x)\}gm(v)$$+\{v^{3}+(2-x)v^{2}+(1-X)v\}g_{m}(\prime v.)$.

If $\deg g_{m}\leq m$ , then $\deg g_{m+1}\leq m+2$ , and if $\deg g_{m}=m+1$ , then $\deg g_{m+}1=m+2$ .

We further see that$g_{m+1}(0)=((l+s-1)/2-m)(1-x)gm(0)\neq 0$ .

The intgral
4$\int_{0}^{\infty}\frac{v^{(l)}/2-n1(+S-3gm+1v)}{(1+v)(\iota_{-}s)/2+2(1+v-X)(\iota+S)/2+m+2}d_{S}$is convergent for$-(l-1)/2+m<\sigma<(\iota+5)/2+m$ .

Thus Theorem is proved by the induction on $m$ .

THEOREM 4.2.

In the strip-M$\leq\sigma\leq M$

口

, we

$have_{j}$$|G(_{S}, \iota;X)|\leq c_{1}(1+\iota+|t|)^{\sigma}-1/2e-\mathrm{C}_{2}|t|$where the positive constants

PROOF.

$c_{1},c_{2}$depend only on $M$ andTo polynomial$P(u)= \sum a_{k}ukk$

$x$

.

,

268

TAKAYOSHI MITSUIof complex coefficients, we put$||P||= \max_{k}(|a_{k}|)$It is obvious that for polynomials $P,$

$Q$

.

,

$||P+Q||\leq||P||+||Q||$ ,$||PQ|| \leq\{\max(\deg P, \deg Q)+1\}||P||||Q||$ ,$|$,$||P’||\leq||P||\deg P$$|P(u)| \leq(\deg P+1)||P||\max(1, |u|^{\deg}P)$.

Now by (3.7),(3.9)$||g0||=1\mathrm{n}\mathrm{a}\mathrm{X}\{(l+1)(1-X), |\iota+1-(l+s)_{X}|\}\leq 1+\iota+|s|$.

By (3.8) we have

(3.10)$||g_{m+1}|| \leq(\max(2, m+1)+1)(l+|s|+2m+2)||g_{m}||$$+ \{(\max(3, m)+1\}2(m+1)||g_{m}||$$\leq 2(m+\mathrm{s})(\iota+|_{S|}+m \dagger)||gm||+(m+4)2(m+1)||gm||$$\leq 4(m+4)(\iota_{+}|_{S1}+m+1)||gm||$.

By (3.9), (3.10) and the induction on $m$ ,(3.11)We denote by$||g_{m}||\leq C(1+l+|S|)^{m+1}$$C$.

the constants depending only on $m$ .

We put$F(v)= \frac{v^{(+)/2-}-1mg\iota\theta m(v)}{(1+v)(l-s)/2+2(1+v-X)(\iota+S)/2+m+1}$,

where we assume(3.12)$-(l+1)/2+m+\cdot\epsilon\leq\sigma\leq(l+3)/2+m-\epsilon$$(0<\epsilon\backslash /1/4)$

.

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MULTIPLE SERIES IN A TOTALLY IMAGINARY NUMBER FIELD II

We consider $F(z)$ for complex variable $z(-\pi<\arg z<\pi)$ . Let$R$

be a large number and

if ${\rm Im}(s)=t\geq 0$ ,if ${\rm Im}(S)=t<0$ .

$\pi/6$$\varphi=\{$$-\pi/6$

In -plane, we take aclosed curve$z$$0arrow Rarrow L_{1}L2Rei_{\Psi_{arrow \mathrm{o}}}L_{3}$$L_{1}$

and$L_{3}$

are the straight lines from

$0$

to

part of the circle with center $z=0$ . On

$L_{2}$$R$

and from.

$Re^{i\varphi}$

to

$0$

respectively, and$L_{2}$: $z=Re^{i\theta}$ , we have, for (3.11) and (3.12),$|F(Re^{i\theta})|\ll R^{()/2-}\sigma-\iota-5m\ll R-1-\epsilon$

.

Hence
$(Rarrow\infty)$$\int_{L_{2}}F(z)dz=\int_{\theta}^{\varphi}=0F(Re^{i\theta})d(Re^{i\theta})\ll R^{-\epsilon}arrow 0$,

which gives$I= \int_{0}^{\infty}F(v)dv=\int_{V}F(z)d_{\mathcal{Z}}$

where

$V$

,

is the half fine$z=ve^{i\varphi}$$.(0\leq v<\infty)$.

We take the absolute value of $F(z)$ :(3.13)$|F(z)|= \frac{|z|^{()}l+\sigma-1/2-m|g_{m}(Z)|}{|1+z|(\iota-\sigma)/2+2|1+z-X|(l+\sigma)/2+m+1}\mathrm{e}\mathrm{x}’\mathrm{p}(-\frac{t}{2}\arg\frac{z(1+Z)}{1+z-x})$

On the line$V$

(3.14)$\arg\frac{z(1+Z)}{1+z-x}=\varphi+\tan^{-1}(\frac{v\sin\varphi}{1+v\cos\varphi})-\tan^{-1}(\frac{v\sin\varphi}{1-x+v\mathrm{c}\mathrm{o}s\varphi})$$= \varphi-\tan^{-1}(\frac{xv\mathrm{s}i\mathrm{n}\varphi}{1-x+(2-x)v\cos\varphi+v^{2}})$

which shows that$\frac{t}{2}\arg\frac{z(1+Z)}{1+z-x}=|\frac{t}{2}\arg\frac{z(1+Z)}{1+z-x}|$

.

,

.

is a

270

TAKAYOSHI MITSUIFor $0\leq v<\infty$ , the function$f(v)= \frac{xv}{1-x+(2-X)v\cos\varphi+v2}$

takes extremal value at $v=\sqrt{1-x}$ . Hence$f(v) \leq\frac{x\sqrt{1-x}}{2-2x+(2-X)\sqrt{1-x}\cos\varphi}=\frac{x}{2\sqrt{1-x}+(2-x)\cos\varphi}$$\leq\frac{1}{\sqrt{1-x}+\cos\varphi}$and by (3.14)(3.15)$| \arg\frac{z(1+z)}{1+z-x}|\geq|\varphi-\tan-1(\frac{\mathrm{s}\mathrm{i}\mathrm{i}1\varphi}{\sqrt{1-x}+\cos\varphi})|=|\tan^{-1}(\frac{\sqrt{1-x}\sin\varphi}{1+\sqrt{1-x}\cos\varphi})|$$\geq\frac{1}{2}\sqrt{1-x}|\sin\varphi|$.

From (3.11), (3.13) and (3.15) we obtain(3.16)$|I| \leq\int_{0}^{\infty}\frac{|z|^{(+}l\sigma-1)/2-m|gm(_{Z})|}{|1+z|(l-\sigma)/2+2|1+z-X|(\iota+\sigma)/2+m+1}e^{-\mathrm{c}|t|}|dz|$$\leq C(1+\iota+|S|)m+1e-c|t|\int_{0}^{\infty}\frac{v^{(\iota)}+\sigma-1/2-m\max(1,v)^{m}+1}{((1-X)\cos\varphi+v)(l+\sigma)/2+m+1}$$\cross\max\{(1+v\cos\varphi)^{(-l)/}\sigma 2-2,$ $(1+v)^{(\sigma-\iota)/2-2}\}dv$.

We divide the last integral of (3.16) into two parts:$\int_{0}^{\infty}=\int_{0}^{1}+\int^{\infty}1=I1+I2$.

Since $(\sigma-l)/2-2\leq m-1/2-\epsilon$ by (3.12),$I_{1} \leq C\int_{0}^{1}\frac{v^{(1}l+\sigma-)/2-m}{((1-X)\cos\varphi+v)(l+\sigma)/2+m+1}dv$.

$=C \int_{0}^{1}(\frac{v}{(1-x)\cos\varphi+v})^{(+1}\mathrm{t}+\sigma)/2-m-\epsilon/2\frac{dv}{v^{1-\epsilon/2}((1-x)\cos\varphi+v)^{2+/\epsilon}m12+/2}$

271

MULTIPLE SERIES IN A TOTALLY IMAGINARY NUMBER FIELD II

Further (3.12) gives $(l+\sigma+1)/2-m-\epsilon/2\underline{>}\epsilon/2$ and consequently we have.

(3.17)$I_{1} \leq C\int_{0}^{1}\frac{dv}{v^{1-\epsilon/2}(1+v)^{2+/\epsilon}m12+/2}\leq C$

If $(\sigma-l)/2-2\leq 0$ , then
(3.18)$I_{2}= \int_{1}^{\infty}\frac{v^{()}l+\sigma+1/2}{(1+v\cos\varphi)(l-\sigma)/2+2((1-x)\cos\varphi+v)^{(+}l\sigma)/2+m+1}dv$$\leq C\int_{1}^{\infty}\frac{dv}{(1+v\cos\varphi)^{()/+/2}\iota_{-}\sigma 2+2((1-x)c\mathrm{o}\mathrm{s}\varphi+v)m1}$$=C1^{\infty}.( \frac{1+vc\mathrm{o}\mathrm{s}_{\Psi}}{(1-x)\mathrm{c}\mathrm{o}\mathrm{s}\cdot\varphi+v})^{m+1/2}\frac{dv}{(1+vc\mathrm{o}\mathrm{s}\Psi)(l-\sigma+5)/2+m}$$\leq C\int_{1}^{\infty}\frac{dv}{(1+v\cos\varphi)^{(5)}\iota-\sigma+/2+m}$$\leq C\int_{1}^{\infty}.\frac{dv}{(1+v\cos\varphi)1+\epsilon}dv\leq C$.

If $(\sigma-l)/2-2>0$ , then

(3.19)$I_{2}= \int_{1}^{\infty}\frac{v^{(\iota\dagger+)/}\sigma 12(1+v)^{(-l)/}\sigma 2-2}{((1-X)\cos\varphi+v)(\iota+\sigma)/2+m+1}dv$$\leq C\int_{1}^{\infty}\frac{(1+v)m-1/2-\epsilon}{((1-X)\cos\varphi+v)m+1/2}dv$$\leq C\int_{1}^{\infty}\frac{dv}{(1+v)^{1+\epsilon}}dv\leq c$

because of $(\sigma-l)/2-2\leq m-1/2+\epsilon$ by (3.12).
Thus we have, by (3.16), (3.17), (3.18) and (3.19),$|I|\leq C(1+^{\iota}+|S|)^{m}+1-e\mathrm{c}\mathrm{I}t|$and(3.20)$|G(s, l;x)|\leq c$$\frac{\Gamma(\frac{s+l}{2})}{\Gamma(\frac{l-S+3}{2}+m)}$$(1+l+|S|)^{m}+1-ec\mathrm{I}t|$Supposing(3.21)$m-1/2+\epsilon\leq\sigma/2\leq m+3/2+\epsilon$

,

.

272

TAKAYOSHI MITSUI

we shall show that

(3.22)$\frac{\Gamma(\frac{s+l}{2})}{\Gamma(\frac{l-s+3}{2}+m)}$$\overline{\perp}\overline{\mathrm{f}}m=\hat{\cup}$, then ($3/.\hat{\Delta}2_{\grave{J}}$$|\Gamma(Z)|=|\Gamma(\overline{z})|$(3.23)

is irue$\leq C(1+\iota+|S|)\sigma-m-3/2$.

$(\lfloor^{\angle\pm}’\Gamma A, (3/.32^{r1}i)\grave{J}\rfloor\grave{J}\cdot \mathrm{A}\mathrm{S}\mathrm{b}\mathrm{t}\iota 11\dot{1}\mathrm{c}r1\overline{\mathrm{b}}1^{-}\perp\overline{\mathrm{d}}^{\mathrm{J}}\mathrm{b}j_{l}^{\wedge}\hat{t}\underline{/\backslash }1$.

$\mathrm{T}.\mathrm{T}\mathrm{s}_{1}^{\neg}\mathrm{n}\mathrm{g}\perp 1_{\wedge}\cup 11\wedge.\mathrm{r}_{\overline{u}\vee}\mathrm{A}-\mathrm{t}$, we write$| \frac{\Gamma(\frac{s+l}{2})}{\Gamma(\frac{l-s+3}{2}+m)}|=\frac{2}{|s+\iota|}|\frac{\Gamma(\frac{s+l}{2}+1)}{\Gamma(\frac{l+s+3}{2}+m-\sigma)}|$ ‘

The difference of the arguments of$\Gamma$-function in the right-hand side is $\{(l+s+3)/2+$$m-\sigma\}-\{(s+l)/2+1\}=1/2+m-\sigma$ , and by the assumption (3.21),$-5/2-m+2\epsilon\leq 1/2+m-\sigma\leq-1/2-m-2\epsilon$.

so we put
$1/2+m-\sigma=-m0^{-}\delta$( $m_{0}=m+1$ or $m+2;|\delta|\leq 1/2$ )and we see(3.24)$| \frac{\Gamma(\frac{l+s+3}{2}+m-\sigma)}{\Gamma(\frac{s+l}{2}+1)}|=|\frac{\tau(z-m_{0^{-}}\delta)}{\Gamma(z)}|$$= \frac{1}{|(z-m_{0^{-}}\delta)\cdots(z-1-\delta)|}|\frac{\Gamma(z-\delta)}{\Gamma(z)}|$where $z=(s+l)/2+1$ . Since${\rm Re}(Z)=(\sigma+l)/2+1\geq(\iota-1)/2+m+1\geq 3/2$ ,we have$C_{1}|z|^{-} \delta\leq|\frac{\Gamma(z-\delta)}{\Gamma(z)}|\leq \mathit{0}_{2}|_{Z}|-\delta$$|\delta|\leq 1/2$

,

,

273

MULTIPLE SERIES IN A TOTALLY IMAGINARY NUMBER FIELD II([4, Hilfssatz 4.17]). Hence it follows from (3.24) that$| \frac{\Gamma(\frac{s+l}{2}+1)}{\Gamma(\frac{l+s+3}{2}+m-\sigma)}|\leq C(1+l+|S|)^{m0}|z|\delta=C(1+\iota+|s|)^{m+\delta}0$

and by (3.23)(3.25)$\frac{l^{\urcorner}(\frac{s+l}{2})}{\Gamma(\frac{l-s+3}{2}+m)}$$\leq C(1+l+|_{S}|)m\mathrm{o}+\delta-1=C(1+\iota+|S|)\sigma-m-3/2$

By this result (3.25) and (3.20),(3.26)$|G(_{S}, \iota;X)|\leq C(1+l+|s|)\sigma-1/2e-c|t|$for$2m+\epsilon\leq\sigma\leq 2m+3-\epsilon$.

If$-2-2m+\epsilon\leq\sigma\leq 1-2m-\epsilon$,

then, using (3.4) and above result (3.26),$|G(_{S,\iota;}X)|=| \frac{\Gamma(s+l)}{\Gamma(1-s+l)}|2^{1-2\sigma}|G(1-S, l ; x)|$$\leq C|\frac{\Gamma(_{S+^{\iota}})}{\Gamma(1-s+\iota)}|(1+l+|s|)^{1/-}2\sigma-ec|t|$

In the same way as is obtained (3.25), we have$| \frac{\Gamma(s+l)}{\Gamma(1-s+l)}|\leq C(1+l+\}_{S}|)2\sigma-1$.

’Thus we have$|G(_{S}, \iota;X)|\leq C(1+l+|_{S|})\sigma-1/2e-\mathrm{C}|t|$and the proof is completed

口\S 5. Representations by integrals

.

.

274

TAKAYOSHI MITSUI

We observe the sum overin (1.6). This sum becomes double sum:$\nu_{I}$(4.1)$\sum_{0\neq\nu_{I}\in \mathfrak{a}I}=\sum_{\nu_{I}/U^{P_{I}}}\sum_{\epsilon\in U^{P}I}$.

$0\neq\nu_{I}\in\alpha_{t}$In this right-hand side, the outer sum is taken over all non-zero numbers

associated to each other with respect to$U^{P_{I}}$$U^{P_{I}}$$\nu_{I}$in$a_{I}$and the inner sum is taken over all unitsnot$\epsilon$of.
The unit$\epsilon$of$U^{P_{I}}$is of the form:$\epsilon=(\rho^{b}\epsilon_{11}^{b..\cdot P_{I}}\epsilon_{\Gamma-}^{b}-1)1.’$

where$b_{1},$$\ldots,$$b_{r-1}$

are integers,$b$,takes the values $b=1,$$\ldots,$$w/d_{I}(d_{I}=(w, P_{I}))$. We seethat$[U : UP_{I}]=d_{I}P_{I}r-1$

We consider the sum over$\epsilon$.

in (4.1) and write it as follows;(4.2)$\sum_{\epsilon\in U^{P}\tau}\exp\{-2P\pi s(\tau\prod_{J}|\nu_{J}|^{1/P_{J}}|\epsilon|^{1/P_{\tau}})+2P\pi iS(\xi\prod_{J}\nu_{J}1/P_{J\epsilon^{1}}/P_{I})\}$$= \sum_{1b=b1b}^{w/I},\ldots,\sum_{\infty\gamma-1=-}\exp\{d\infty-2P\pi s(T\prod_{J}|\nu J|^{1}/P|J\epsilon_{11}\ldots \mathcal{E}_{r-}1b1b_{r-}|)$$+2P \pi iS(\xi\prod\nu_{J}\rho^{b}\epsilon_{1}\cdot\cdot\epsilon_{r}^{b_{r_{1}}}--1/Pb_{1}.1j)J\}$.

Now we quote the following theorem from [4, Hilfssatz 14]:
THEOREM 5.1.
$1,$ $\ldots,r)$

. Let

Let

$U_{1,)}\ldots U_{n}$$W_{1}\ldots.,$ $W_{n}$be positive numbers such thai $W_{p+r}=W_{p}(p=$be complex numbers such that $U_{p+r}=\overline{U}_{p}(p=1, . . ., r)$ . Then

275

MULTIPLE$\dot{\mathrm{S}}$ERIES IN A TOTALLY IMAGINARY NUMBER FIELD II

we have$b_{1}, \ldots,b_{1-}.1=\sum_{-\infty}^{\infty}\exp\{$ $-2\pi

S(W|6_{1r-}^{b..b,.b_{-1},\}}1.\epsilon|-11)+2\pi iS(U_{\mathcal{E}^{b}}1\ldots\epsilon 1r-1)$$\sum\infty$$= \frac{2^{r}}{R}$$\sum\infty$$\frac{1}{2\pi i}\int_{(\sigma 0)}(2\pi)^{-nS}$$m_{1},\ldots,m,.-1=-\infty\infty$$\infty\infty$$a_{1},\ldots,a_{n}\geq 0$$a_{p}\cdot a_{\mathrm{p}+\gamma^{=0}}$;$\cross\prod_{p=1}^{r}.\{\frac{(iU_{p})^{a_{p}}(iU_{p}+r)a_{p+}t}{(W_{p}^{2}+|U|^{2}p)^{(}2\theta+ivp+l_{P})/2}G(2s+iv\iota_{p}$$\frac{|U_{p}|^{2}}{W_{p}^{2}+|U|^{2}p}p’)\}ds$where$m_{1},$ $\ldots,$$m_{r-1}$

run throught all rational in tergers,$a_{1},$$\ldots,$$a_{n}\mathrm{r}u\mathrm{n}$tive rational integers such that $a_{p}\cdot a_{p+r}=0(p=1, \ldots, r)$ . The$v_{p}$,

through non-nega-

are the values definedby $(2,1)$ and $G(s, l;X)$ is the function defined in \S 4. We put $l_{p}=a_{p}+a_{p+r}(p=1, \ldots, r)$and the integrals are the complex integrals taken along the vertical line$\sigma=\sigma_{0}$.

Applying this Theorem to the sum in the right-hand side of (4.2), we have
(4.3)
$6 \in U^{P}\sum_{t}\exp\{.-2P\pi S(\mathcal{T}\prod_{J}|_{\mathcal{U}}J|^{1/}P_{J}|\epsilon|1/P_{t})+2P\pi iS(\xi\prod_{J}\nu_{J}^{1}\mathcal{E}I/P_{J}1/P)\}$$= \frac{2^{r}}{R}\sum\sum w/d_{tn}q)^{ab}q()(2P\pi$$b=1\{m\}\{a\}q=1$$\cross\prod_{p=1(\tau_{p}\prod|\nu(p)|^{1}J)2+}^{r}\{JJ|\xi_{p}\prod jJ|\nu\}^{()/2}(i\xi \mathrm{P}\prod_{/P}JJ)^{a}\nu^{(}p)^{1}/P_{J}p(p)22s+iv_{p}+\iota a_{p+}tp$$l;p \mathrm{P}\frac{|\xi_{p}|^{2}}{\tau_{p}^{2}+|\xi_{p}|^{2}})dS$$\cross\prod_{p=1}^{r}G(2S+iv,$

where we denote by$\sum_{\{m\}}$and$\sum_{\{a\}}^{*}$,

the summations over$m_{1},$$\ldots$,$m_{r-1}$

and$a_{1)}\ldots$,$a_{n}$

inthe meaning of Theorem 5.1, respctively, and the integrations are taken along the vertical
line
(4.4)

$\sigma_{0}=\max\{(\iota-\alpha_{J})P_{j}\}+1$$J\in J$.

276

TAKAYOSHI MITSUI

Putting this formula (4.3) into (4.2) and then into (1.6), we have
(4.5)

$M(\tau, \xi;\alpha_{J}, a_{J})$$= \sum\cdots\sum$$\frac{2^{r}}{R}\sum\sum^{*}\sum w/d_{I}\prod\rho^{(q)^{a_{q}}}\frac{1}{2\pi i}\int_{(\sigma 0)}b)^{-nS}n(2P\pi$$\sum$.

$0\neq\nu_{J}\in\alpha j0\neq\nu_{I}\in \mathrm{Q}I$$\{m\}\{a\}$$b=1q=1$$\nu_{I}\mathit{1}’\cup\tau P$$\nu_{J}/^{\mathrm{u}}(j\neq’\tau\tau n_{I}’(r\backslash \backslash )’$$\cross\prod_{p=1}^{r}\{\frac{(i\xi_{p})^{a_{\mathrm{p}}}(i\xi_{pr}+)^{a_{\mathrm{p}}}+t}{(\tau_{p}^{2}+|\xi_{p}|^{2})(2S+iv+p\iota_{\mathrm{p}})/2}G(2_{S}+iv_{p’ p}l ; X_{p})\}$$\cross\prod_{J}\{N(\nu_{\int})^{-\alpha}J\prod^{r}\frac{\nu_{J}^{\langle p)}a_{\mathrm{p}}/P_{J}\nu_{J}(p+r)a+r/\mathrm{p}P_{j}}{|\nu_{J}^{(p)}|(2_{S+i+\iota_{\mathrm{p}}}v_{p})/P_{J}}\mathrm{P}=1\}dS$,

where we put$x_{p}= \frac{|\xi_{p}|^{2}}{\tau_{p}^{2}+|\xi \mathrm{P}|^{2}}$$(p=1, \ldots, r)$.

By Theorem 42, we can estimate the integrand in the right-hand side of (4.5):$(2P \pi)^{-}ns\prod_{p=1}^{r}\{\frac{(i\xi_{\mathrm{P}})a_{\mathrm{p}}(i\xi_{p}+r)ap+\tau}{(\tau_{p}^{2}+|\xi p|^{2})^{(\iota}2_{S+v+)}iP\mathrm{p}/2}G(2S+ivl;\mathrm{P}x)\mathrm{P}’ \mathrm{P}\mathrm{I}$$\cross\prod_{J}\{N(_{\mathcal{U}}J)^{-\alpha}J\prod^{r}\frac{\nu_{J}^{(p)^{a_{p}}}\mathcal{U}_{J}/P_{J(}p+r)a_{p}+r/P_{J}}{|\nu_{J}|^{(+}(p)2Siv\dagger pl)\mathrm{p}/P_{J}}\}\mathrm{P}=1$$\ll\prod_{J}N(l^{\text{ノ}}J)^{-}\alpha-j\sigma_{0}/P_{J}\prod_{p=1}^{r}x_{\mathrm{P}}^{\iota/2}p(1+l_{\mathrm{P}}+|2t+v|)22\sigma 0-1/2ep-\mathrm{c}|t+v_{\mathrm{p}}|$

.

Hence
$M(\tau, \xi\alpha_{J}, \alpha_{J})$$\ll\sum\cdots\sum$$\sum$$\prod N(\nu_{J})-\alpha_{J^{\wedge}}\sigma 0/P_{j}$$0\neq\nu\epsilon ja\nu_{J}/U^{\psi}(J)J0\neq\nu\nu_{I}/U^{P_{t}}I\in a_{I}J$$(J\neq I)$$\cross\sum\sum^{*}\prod rx^{\mathit{1}_{p}/2}\int p\prod_{p=1}(1+\iota_{p}+|2t+v_{p}-\infty\infty|)r2\sigma 0^{-}1/2-e\mathrm{c}|2t+v|pdt$$(m\}\{a\}p=1$

In this right-hand side,$\sum\sum^{*}\prod x^{l/2}rp\nu\int_{-}\prod_{\infty=1}^{\infty}(1\mathrm{p}tr+l_{p}+|2+v_{p}|)^{2\sigma-1/}02e-\mathrm{C}12t+v_{p}1_{d}t$$\{m\}\{a\rangle p=1$

.

277

MULTIPLE @J.RIES IN A TOTALLY IMAGINARY NUMBER FIELD II

is convergent ([2]). The convergence of$\sum\cdots\sum$$\sum$$\prod N(\nu_{J})^{-\alpha_{j^{-\sigma_{0}}}}/P_{J}$$0\neq\nu_{j}\in a_{J}\nu_{J}/U^{\phi}(J)0\neq\nu t\in\nu_{I}/UIP_{I}\emptyset J$$(J\neq I)$

is clear since$\alpha_{J}+\sigma_{0}/P_{J}\geq\alpha_{J}+((1-\alpha_{J})PJ+1)/P_{J}=1+1/P_{J}$$(J\in J)$

by (4.4).

Thus the right-hand side of (4.5) is absolutely convergent. The order of the summa-

tions and integrations can be exchanged. Consequently we have$M(\tau, \xi;\alpha_{J}, aJ)$

(4.6)$= \frac{2^{r}}{R}\sum\sum^{*}\sum w/dIn\prod\rho^{(q)^{ab}}q\frac{1}{2\pi i}\int_{(\sigma}0))^{-nS}(2P\pi$$b=1q=1$$\{m\}\{a\}$$\cross\prod_{p=1}^{r}\{\frac{(i\xi_{\mathrm{P}})a_{p}(i.\xi_{p+}r)^{a_{p}}+f}{(_{T_{\mathrm{P}}^{2}}+|\xi p|2)(2s+iv_{p}+l_{\mathrm{p}})/2}G(2s+iv\iota_{pp} ; Xp’)\}$$r$$(p)^{a_{p/J}}\Gamma(p+r)ap+r/\Gamma J$$\sum\cdots\sum\prod_{J\mathrm{n}\mathit{4}u.\in\alpha\prime}N(\mathcal{U}_{J})-\alpha_{J}\prod\prod Jp=1\frac{\nu_{J}\nu_{J}}{|\nu^{(p)}j|^{(i}2S+v+pl)p/P_{J}}d_{S}$$\cross\sum_{0\mathit{4}u.G\dot{\mathrm{d}}}$.

$J$$\nu_{\mathit{1}}/PI|\neq\nu_{U}\in t\emptyset I\nu_{J}0\neq\nu_{J}/U\phi\in(J)0j$$(J\neq I)$

In this integrand, the sum over$\{\nu_{J}\}_{J}$is$(4.7)$$\sum$$\sum\cdots\sum\prod_{J}\{N(\nu_{J})^{-\alpha}J^{-}/SPJ\prod_{=p1}r|\nu_{J}|(p)-iv/\mathrm{P}P_{J}\prod_{q=1}^{n}(\frac{\nu_{J}^{(q)}}{|\nu_{J}^{(q)}|})^{a_{q}/P}J\}$$0\neq\nu_{J}\in\alpha J$$0\neq\nu t\in \mathrm{Q}I$$\nu_{I}/U^{P}t$$\nu_{J}/(J\neq I)U\phi(J)$$=$$\sum$$\{N(\nu_{I})-S/PIp=1\prod r|\nu_{I}^{(p)}|-iv_{p}/P_{I}\prod_{q=1}^{n}(\frac{\nu_{I}^{(q)}}{|\nu_{I}^{(q)}|})^{a_{q}/P_{t}}\}$$0\neq\nu_{I}\in a_{I}$$\nu_{I}/U^{P}t$$\cross\prod$$\sum$$J\neq I0\neq\nu_{J}\in aJ$$\nu_{J}/U^{\phi(J)}$$\{N(\nu_{J})^{-}s/P_{J}-\alpha j\prod_{1p=}^{\gamma}|\nu_{J}|(p)-iv\mathrm{P}/p_{J}\prod_{q=1}^{n}(\frac{\nu_{J}^{(q)}}{|\nu_{J}^{(q)}|}\mathrm{I}^{a_{q}/P_{J}}\}\cdot$

278

TAKAYOSHI MITSUIFirst we consider the um over
$s$

(4.8)

$\nu_{I}$and write it as follows:$\sum_{0\neq\nu\in ta_{I}}=\sum_{0\neq\nu I\in\emptyset_{I}}$$\nu_{I}/U$$\nu_{I}/U^{P}I$$\cross\prod_{q=1}^{n}(\frac{\overline{6}^{\backslash A}\prime\overline{\iota}\text{ノ_{}I}\prime_{\theta})(q)}{|\epsilon^{(q)}\nu_{I}|(q)})^{a_{q}}\}/\backslash /PI\backslash .-$In the inner sum,$\epsilon$

runs through the representatives of$\epsilon=\rho^{bb_{1}.’1}\epsilon_{1}\cdot\cdot\epsilon^{b}r’-1-$$U/U^{P_{I}}$. Hence we can put

,

where
$b=0,$and the sum over$\epsilon$$\ldots,$$d_{I}-1$,$b_{j}=0,$ $\ldots,$ $P_{I}-1$$(j=1, \ldots, r-1)$

in the right-hand side of (4.8) is(4.9)$\sum_{\epsilon\in U/U^{P}I}p=1\prod r|\epsilon^{(}|^{-iv/P}p)pt\prod_{q=1}^{n}(\frac{\epsilon^{(q)}}{|\epsilon^{(q)}|})^{a_{q}/P_{I}}$$= \sum_{b=0}^{d_{I}-1}\prod_{q=1}^{n}\rho^{(}\sum_{b_{1}0}^{x}q)ba_{q}/PtP=-1$...

$\sum_{b_{r-1}0}^{P_{t}1}-=\prod_{p=1}^{r}|\epsilon_{1}^{(p)}b_{1}$...

$\epsilon_{r-1}^{(p})^{b}f-1|^{-iv_{p}/P_{I}}$$\cross\prod_{q=1}^{n}\{(\frac{\epsilon_{1}^{(q)}}{|\epsilon_{1}^{(q)}|})^{b_{1}}\cdots(\frac{\epsilon_{r-1}^{(q)}}{|\epsilon_{r-1}^{(q)}|})^{b_{r-1}}\}^{a_{q}/P_{I}}$Since$(p=1, \ldots, r)$$v_{p}= \sum_{j=1}^{r-1}e^{(j)}p(2\pi m_{j}+\sum_{q=1}^{n}a_{q}\arg\epsilon j)(q)$,we have$\sum_{p=1}^{\mathrm{r}}v\mathrm{l}\mathrm{o}\mathrm{P}\mathrm{g}|\epsilon_{j}^{(_{\mathrm{P}})}|=2\pi m_{j}+\sum_{q=1}^{n}a_{q}\arg\epsilon j(q)$and the sum over$b_{1},$$\ldots,$$b_{r-1}$$(j=1, \ldots, r-1)$

in the right-hand side of (4.9) is$P_{I}-1 \sum\ldots P_{I}-1\sum\prod|\epsilon_{11}^{(p)}\epsilon_{r-}r|b\iota\ldots(p)b_{r}-1-:v/pP_{I}$$b_{1}=0$$b_{\gamma-1}=0p=1$

.

279

MULTIPLJ SERIES IN A TOTALLY IMAGINARY NUMBJR FIELD II$\cross\prod_{q=1}^{n}\{(\frac{\epsilon_{1}^{(q)}}{|\epsilon_{1}^{(q)}|})^{b_{1}}$...

$( \frac{\epsilon_{r-1}^{()}q}{|\epsilon_{r-}^{(q)}|1})^{b,-}1\}a_{q}/P_{I}$$= \sum P_{I}-1\ldots P_{I}-1\sum\exp\{-\frac{i}{P_{I}}\sum_{=}^{r}v_{p}\sum^{1}bj\log|\epsilon_{j}^{()}pp1jr-=1|$.

$b_{r-1}=0$$b_{1}=0$$+ \frac{i}{P_{I}}\sum_{1q=}^{n}\sum_{j=}^{1}r-1$$= \sum P_{I}-1$

..

’

b

O qj

$\mathrm{r}\mathrm{g}\epsilon j^{q}()\}$$\mathrm{a}$$P_{I}-1 \sum\exp\{-\frac{2\pi i}{P_{I}}\sum_{1j=}^{-1}m_{j}b_{j\}}r$$b_{r-t=}0$$b_{1}=0$$P_{I}^{r-1}$

$=\{$
$0$

if
otherwise.

$P_{I}|m_{1}\cdot,$$\ldots,$$P_{I}|m_{r-1}$,

Putting this result into (4.9) and then into (4.6), we have

(4.10)$M(\tau, \xi;\alpha J’ a_{J})$.

$= \frac{2^{r}}{R}P_{I}’-$ $1 \sum_{\}\{m}\sum_{\mathrm{t}a\}}\sum_{b=1q}^{t}\prod^{n}’*w/d=1\rho\sum_{=0}(q\rangle a_{q}ba\prod_{q1}^{n}\rho/(q)^{b}qPd_{I}-1b-I$$\cross\frac{1}{2\pi i}\int_{(\sigma_{0})}(2P\pi)-ns\prod_{=p1}^{r}\{\frac{(i\xi_{\mathrm{P}})a_{p}(i\xi_{p}+r)^{a_{p}}+\prime}{(\tau_{p}^{2}+|\xi_{p}|2)(2s+iv_{p}+\iota_{p})/2}G(2_{S}+iv_{p}, \iota;\mathrm{P}X)p\}$$\cross\sum_{0\neq(\nu_{t})\subset a_{x}}\{N(\nu_{I})^{-}s/P\prod I|\nu^{(p}I)|^{-i/P}vp=1rpx\prod_{q=1}^{n}(\frac{\nu_{I}^{(q)}}{|\nu_{I}^{(q)}|})^{a_{q}/P}x\}$$\cross\sum\cdots\sum\{N(\nu_{j})^{-\alpha_{J}-}\theta/P_{j}\prod^{r}|\nu^{(p}|^{-}p=1^{\cdot}.q\prod_{\overline{\sim}1}^{n}J)ivp/PJ(\frac{\nu_{J}^{(q)}}{|\nu_{J}^{(q)}|})^{a_{q}/P}J\}ds$,$0\neq\nu_{J}\in a_{J}$$\nu_{J}/U^{\phi}(j\neq I)(j)$where the sum
multiples of$P_{I}$$\sum_{\{m\}}’$

means that the summation variables$m_{1},$ $\ldots,$ $m_{r-1}$

run through all.

Next we consider the sum over $\nu_{J}(J\neq I)$ in the right-hand side of (4.7). In the same

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TAKAYOSHI MITSUI

way as (4.8), we write$\{N(\nu_{J})^{-}\alpha-JS/P\prod J|\nu_{J}|(p)-iv_{p}/P\prod_{=p=1}^{n}r1Jq(\frac{\nu_{J}^{(q)}}{|\nu_{J}^{(q)}|})^{a_{q}}/P_{j\}}$$\sum$$0\neq\nu_{J}\in a_{J}\nu_{J}/U^{\phi}(J)$$=$$\overline{\sum}$$\{^{(}N(\nu J)^{-}\alpha-J/l^{y}J\overline{\sum_{6}}S\frac{r}{p=111}(_{P}.)\mathcal{U}_{J}|\epsilon|\mathrm{t}\prime P)-\dot{l}v/p.\overline{P}_{j}$$0\neq\nu_{J}\nu_{J}/U\in aJ$$\cross\prod_{q=1}^{n}(\frac{\epsilon^{(q)_{\mathcal{U}_{J}})}(q}{|\epsilon^{(q)}\nu_{j}|(q)}\mathrm{I}^{a_{q}/P_{J}}\}$

where$\epsilon$

runs through the representatives of$U/U^{\phi(J)}$$\epsilon=\rho^{b}\epsilon^{b_{1}}’\cdot\cdot\epsilon_{r-1}^{b_{\mathrm{r}}}1-1$,. Hence we can put

,

where

The sum over$\sum$$\epsilon$,$b=0,$ $\ldots,$ $d_{J}-1$ ,$d_{J}=(w, \phi(J))$$b_{j}=0,$ $\ldots,$ $\phi(J)-1$$(j=1, \ldots, r-1)$ .

is$\prod r|\epsilon^{(p)i}|^{-}v_{\mathrm{P}}/P_{j}\prod_{q=1}^{n}(\frac{\epsilon^{(q)}}{|\epsilon^{(q)}|})^{a_{q}/P_{J}}$$\epsilon\in U/U^{\phi(J})p=1$$=d_{J}-1 \sum\prod\rho^{(q)^{ba_{q}/P_{J}}}n\phi(J)-1\sum\ldots\phi(J)-1\sum\prod^{r}|\epsilon_{1}^{(p)^{b}}\mathrm{t}\ldots\epsilon_{r-1}|^{-i/j}(p)r-v_{\mathrm{P}}b\iota P$$b=0q=1$$b_{1}=0$$b_{r-1}=0_{\mathrm{P}}=1$$\cross\prod_{q=1}^{n}\{(\frac{\epsilon_{1}^{(q\rangle}}{|\epsilon_{1^{q}}^{()}|})^{b_{1}}$In this right-hand side, the sum over$b_{1},$$\ldots,$$b_{r-1}$...

$( \frac{\epsilon_{r-1}^{(q)}}{|\epsilon_{r-1}^{(q)}|})^{b_{t}}-1\}^{/P_{j}}aq$is.

$\phi(J)b_{1}=\sum_{0}^{-1}\cdots\emptyset b_{r}\sum_{-\iota 0}^{(}\exp\{-\frac{2\pi i}{P_{J}}\sum_{j=1}^{-}m_{j}b_{i\}}J)-1=r1$

.

281

MULTIPLE$\mathrm{S}J$

Since we can replace all(4.11)RIES IN A TOTALLY IMAGINARY NUMBER FIJLD II$m_{j}$

by$P_{I}m_{j}$, this sum is$. \emptyset()\sum_{b,-1=0}^{J}-1\exp\{-2\pi i\frac{P_{I}}{P_{J}}\sum_{j=1}mr-1jbj\}$$\emptyset(J\rangle-b\iota\sum_{=0}^{1}\cdot\cdot$$\phi(J)^{r-1}$

$=\{$
$0$

if $\phi(J)|m_{1},$otherwise,$\ldots,$$\phi(J)|m_{r-1}$,

because of $\phi(J)=P_{J}/(P_{I}, P_{J})((1.1))$ .

We see that.

$P_{I}1_{\mathrm{C}}\mathrm{m}_{I}J\neq J\in\{\phi(J)\}=1\mathrm{c}\mathrm{m}\{JP_{J}\}=L$

Hence (4.10) and (4.11) show that
$M(\tau, \xi;\alpha_{J}, a_{J})$$= \frac{2^{r}}{R}P_{I^{-}}^{\Gamma}1\prod_{J}\phi(J)r-1\sum_{\}i^{m}}\sum_{a}\prime\prime \mathrm{t}1*w/\sum_{1b=}^{d}\prod_{1q=}^{n}\rho)I(qabq\prod_{J}\sum_{b=0}^{d-1}jq=\prod_{1}^{n}\rho)(qba_{q}/P_{J}$$\cross\frac{1}{2\pi i}\int_{(\sigma_{0})}(2P\pi)-ns\prod_{=p1}^{r}\{.\frac{(i\xi_{p})^{a}\mathrm{p}(i\xi_{\mathrm{P}+r})a_{P+}r}{(\tau_{p}^{2}+|\xi p|^{2})(2s+iv_{pp}+\iota)/2}G(2s+iv_{p}, l;pxp)\}$,$\cross\prod_{Jy}\sum_{0\neq(j)\subset aJ}\{N(\nu j)^{-S}/P_{J^{-}}\alpha\prod_{p1}J|_{\mathcal{U}}(J|\mathrm{P})-iv_{p}/P_{j}\prod^{n}=rq=1(\frac{\nu_{J}^{(q)}}{|\nu_{J}^{(q)}|})^{a_{q}/P}J\}dS$

where the sum$\sum_{\mathrm{t}’m\}}’$

means that the summation variables$m_{1},$ $\ldots,$$m_{r-1}$

run through all

multiples of .
$L$$\mathrm{R}\mathrm{o}m$this expression we easily obtain$T( \tau, \xi;\alpha J’ aJ)=\sum^{-}hL1=01$:..

$h_{f} \sum_{0=}^{L-1}.M(\tau, \xi_{1}\omega, \ldots, \xi_{r}\omega^{h_{r}} ; \alpha_{J}, a_{J})h_{1}$$= \frac{2^{r}}{R}P_{I}^{r}-1\prod\emptyset(JJ)r\cdot-1\sum_{\}\mathrm{t}^{m}}.\sum_{\}\{ab}\sum_{=}^{d_{I}}\prime\prime*w/1q1\prod_{=}^{n}\rho^{(q})^{ab}q\prod\sum JdJ-1b=0q\prod^{n}\rho^{()^{b}}=1qa_{q}/P_{J}$$\cross\frac{1}{2\pi i}\int_{(\sigma_{0})}(2P\pi)^{-nS}$$\cross\sum_{h_{1}0}^{1}L-=$...

$h,= \sum_{0p}^{L-1}\prod_{1=}^{r}\{\frac{(i\xi_{p}\omega^{h_{p}})ap(i\xi_{pr}+\overline{\omega}^{h}p)^{a_{p}}+r}{(\tau_{p}^{2}+|\xi p|2)(2S+ivp+\iota_{\mathrm{p}})/2}G(2_{S}+iv\iota;X_{\mathrm{P}})p’ p\}$$\cross\prod_{J0\neq}\sum_{\subset(\mu)J\emptyset J}\{N(\nu j)^{-s}/PJ^{-\alpha}Jp1\prod_{=}r|\nu_{J}^{(_{\mathrm{P}})}|-iv/pP_{J}\prod_{q=1}^{n}(\frac{\nu_{J}^{(q)}}{|\nu_{J}^{(q)}|})^{a_{q}/P_{J}}\}ds$

,

282

TAKAYOSHI MITSUI

Since
$L-1$

$L-1$

$L-1$

$r$

$\sum_{h_{1}=0}\cdots\sum\prod\omega^{h_{\mathrm{P}}}(a_{\mathrm{P}}-a+r)p=\sum$$\sum\prod\exp(2\pi i\frac{h_{p}}{L}(a_{\mathrm{P}}-a_{p+r}))$$\ldots,$$0$$m_{1},$ $\ldots,$$r$$h,=0p=1$

if $L|a_{1},$otherwise,$L^{r}$$=\{$

we have, replacing all...

$h_{1}=0$$h_{r}=0_{\mathrm{P}^{=}}1$$L-1$$m_{r-1},$ $a_{1},$ $\ldots,$ $a_{n}$$L|a_{n}$,by $Lm_{1)}\ldots,$ $Lmr-1$)$La_{1},$ $\ldots,\dot{L}an$’(4.12)$T(\tau, \xi;\alpha J’\alpha J)$$= \frac{2^{r}}{R}\{P_{I}\prod_{J}\emptyset(j)\}r-1Lr\sum_{\{m\}}\sum\sum^{w/}*\prod p\mathrm{t}a\}b=0d_{I}q=1n(q)La_{q}b\prod Jd\sum_{b=0}^{J}\prod\rho-1q=1n(q)bLa/qP_{J}$$\cross\frac{1}{2\pi i}\int_{(\sigma_{0}})=\frac{(i\xi_{p})La_{p}(i\xi_{pr}+)^{La}p+}{(T_{p}^{2}+|\xi_{p}|2)^{(2}S+iLv_{p}+L\iota)p/2}(2P\pi)^{-}ns\prod_{p1}’ G(2s+iLv_{p}, Ll_{p} ; xr.)p$$\cross\prod_{J0\neq(\nu_{j})}\sum_{J}\subset\emptyset\{N(\mathcal{U}_{j})-\alpha\mapsto\theta/P,\prod_{1}|p=\nu_{j^{\mathrm{P})/P}}r(|^{-}iLvpJ\prod_{q=1}^{n}(\frac{\nu_{J}^{(q)}}{|\nu_{J}^{(q)}|})^{La_{q}}/Pd_{S}j\}\cdot$

Now consider$\sum_{b=1}^{w/}\prod_{=q1}p^{(}\prod_{J}dxnq)Lab\mathrm{f}dj-1\sum_{b=0}\prod_{=q1}pn(q)bLaq/P_{j}$.

Since $d_{I}|L$ , so$(_{q=1} \prod^{n}\rho^{(q}))^{La_{q}}w/dt=1$

.

Hence
(4.13)$w/d_{I}$$w/I \sum_{b=1q}^{d}\prod^{n}p^{(}=1q)La_{q}b=\{$$0$

if $\prod_{q=1}^{n}\rho(q)La_{q}=1$ ,if not.

This first case occurs if and only if$(_{q=1} \prod^{n}\rho^{(}q)a_{q})(w,L)=1$

Assuming this condition, we consider the sum$d_{I}-1 \sum\prod\rho^{(q)^{L}}naqb/Pl$$b=0q=1$

.

.

283

MULTIPLE

Since $(w, L)$ divides$\mathrm{S}J$

RI$d_{I}L/P_{I}$$J\mathrm{S}$IN A TOTALLY IMAGINARY NUMBER FIELD II

, so

.

$(_{q=} \prod_{1}^{n}p^{()^{La}q}q/P)^{d_{I}}I=1$

Hence.
(4.14)

if $\prod_{q=1}^{n}\rho^{()}qLa/qP_{I}=1$ ,if not.

$d_{I}$$\sum_{b=0}^{d_{I}}\prod_{=q1}\rho-1n(q)Laqb/P_{t}=\{$$0$This first case occurs if and only if$(_{q=1} \prod^{n}\rho^{(}.)^{(/P)}q)a1qw,LI=$.

Under this condition we consider$\sum_{b=0}^{d_{J}-1}\prod\rho)(qqq=1nbaL/P_{J}$$(J_{\overline{7}}\leq_{I)}$

.

Since
$\frac{L}{P_{J}}\frac{d_{J}}{(w,L/P_{I})}.=\frac{L(w,\phi(J))}{P_{J}(w,L/P_{I})}=\frac{(wL,L\phi(J))}{(wP_{J},LP/JP_{I})}$is an integer, $(w, L/P_{I})$ divides $Ld_{J}/P_{J}$ and$(_{q=1} \prod^{n}\rho(q)La/qPJ\mathrm{I}^{d_{J}}=1$

.

Hence
(4.15)$d_{J}$$. \sum_{0b=}^{d_{\Gamma}-1}\prod_{=q1}pn(q)Lba_{q}/P_{J}=\{$$0$

if $\prod_{q=1}^{n}\rho^{()}qLa/qP_{j}=1$ ,if not.

By (4.14) and (4.15), we have, under the condition$\prod_{q=1}^{n}\rho^{()^{La}q}q=1$,

that(4.16)$\prod_{Jb}\sum_{=0q}^{d_{J}-1}\prod_{1=}^{n}p(q)bLa_{q}/P_{J}$$La_{9}/PJ$$\prod_{J}d_{J}$

$=\{$
$0$

)
if $\prod_{q=1}^{n}\rho^{(}=1q$
for all $J\in J$ ,otherwise.

284

TAKAYOSHI MITSUI

Since$L=1\mathrm{c}\mathrm{m}_{J}\{P\}J$’

we see that$\mathrm{g}\mathrm{c}\mathrm{d}_{J}\{L/P_{J}\}=1$. Hence there exist integers$\{n_{J}\}_{J}$that$\sum_{J}??J^{\frac{L}{P_{J}}=1}$ ’which shows that the first conditon of (4.16) is equivalent to a single form$\prod_{q=1}^{n}\rho^{(}q)^{a_{q}}=1$.

Consequently the products$\lambda(\nu_{J})=\prod_{=p1}^{r}|_{\mathcal{U}_{J}}(p)|^{-}iv_{\mathrm{p}}\prod_{q=1}n(\frac{\nu_{J}^{(q)}}{|\nu_{J}^{(q)}|})^{a_{q}}$in the integrands of (4.12) define the Gr\"ossencharacters .
$\lambda$

Thus we have from (4.12), (4.13) and (4.16)(4.17)$T( \tau, \xi;\alpha_{J’ J}a)=\frac{2^{r}w}{Rd_{I}}\{P_{I}\prod\emptyset j(J)\}^{r}-1L^{\Gamma}\prod_{J}dJ$$\cross\sum_{\lambda}\frac{1}{2\pi i}\int_{(\sigma_{0})}(2P\pi)^{-nS}$$\cross\prod_{p=1}^{r}\{\frac{(i\xi_{p})^{La_{p}}(i\xi_{pr}+)^{L}a_{\mathrm{p}}+r}{(\tau_{p}^{2}+|\xi_{p}|2)(2S+iLvp+Ll_{p})/2}G(2S+iLv_{p’ pp}L\iota ; X)\}$$\lambda^{L/}PJ$$\cross\prod_{J}\zeta(\frac{s}{P_{J}}+\alpha_{J},$where the sum$\sum_{\lambda}$

We denote by; $a_{J})ds$ ,is taken over all Gr\"ossencharacters$H_{\lambda}$($s,$ $\tau,$ $\xi;\alpha_{J}$$\lambda$.

, a ) the integrand of (4.17):$J$(4.18)$H_{\lambda}(_{S,\mathcal{T}}, \xi;\alpha J’ a_{j})$$=.(2P \pi)^{-}ns\prod_{1p=}\{r\frac{(i\xi_{p})^{La}p(i\xi_{p+\Gamma})^{La}p+}{(_{\mathcal{T}_{\mathrm{P}}^{2}}+|\xi_{p}|^{2})^{(L}2s+iLv_{P}+\iota_{\mathrm{p}})/2},.G(2S+iLv_{\mathrm{P}}, L\iota_{p} ; X)p\}$$\lambda^{L/}P_{j}$$\mathrm{x}\prod_{J}\zeta(\frac{s}{P_{J}}+\alpha_{J},$;$a_{J})$$=(2P \pi)^{-}n\epsilon\prod_{1p=}\{r\frac{(\mathrm{i}\xi_{p})^{La}p(i\xi_{p+}r)La_{p}+\prime}{(_{\mathcal{T}_{p}^{2}+}|\xi_{p}|^{2})^{(L+}2S+iv_{\mathrm{p}}Ll_{\mathrm{p}})/2}$

such

285

MULTIPLE$\mathrm{S},\mathrm{E}\mathrm{R}\mathrm{I}J\mathrm{S}$IN A TOTALLY IMAGINARY NUMBER FIELD II$\cross F(2s+iLv_{p}, L\iota p;x_{p})\}$$\cross\prod_{J}((\frac{s}{P_{J}}+\alpha_{J},$$\lambda^{L}/PJ$;$a_{J})$and put$A= \frac{2^{r}w}{Rd_{I}}\{P_{I}\prod_{J}\emptyset(J)1^{r-}1L^{r}\prod_{J}d_{J}$.

In particular, we have
(4.19)$H_{1}(S, \mathcal{T}, \xi;\alpha J’ aj)$$= \frac{\Gamma(2S)^{r}}{(2^{2}P\pi)^{ns}}\prod_{p=1}^{r}\{\frac{1}{(\tau_{\mathrm{p}}^{2}+|\xi_{p}|^{2})^{\mathit{8}}}F(s,$$\frac{1}{2}-s,$$1$; $xp) \}\prod_{J}\zeta(\frac{s}{P_{J}}+\alpha J’ aJ)$ .

\S 6. Properties of integrands
THEOREM 6.1.
(2)

If$\lambda=1$(1), then$H_{1}$

If($\lambda\neq 1$$s,$ $\tau,$, then$\xi;\alpha_{J}$$H_{\lambda}$($s,$ $\tau,$ $\xi;\alpha_{J}$, a ) is an eniire function of .
$s$

$J$

, a ) has simple poles at$J$$s=S_{J}^{0}=-\alpha {}_{J}PJ$$(J\in J)$

and$s=s_{j}=(1-\alpha_{J})PJ$

PROOF.

$(J\in J)$.

First we shall show that

(5.1)

$p= \prod_{1}^{r}\Gamma(2s+iLv_{p}+Ll)p=g(S, \lambda)\prod\Gamma J(\frac{s}{P_{J}}+\alpha_{J};\lambda^{L}/P_{J)}$whereis an entire function of$g(s_{)}\lambda)$

$s$

,

.

We write$\prod_{J}\Gamma(\frac{s}{P_{J}}+\alpha J$;$\lambda L/P_{J})=\prod_{p=1}r\prod_{J}\tau(\alpha_{j}^{(k))}+\frac{2s+iLv_{p}+L\iota_{p}}{\phi_{0}\cdots\phi_{k}}$

.

286

TAKAYOSHI MITSUIWe use a well-known formula of the gamma function;(5.2)$\prod_{j=1}^{m}\tau(z+\frac{j-1}{m})=\frac{(2\pi)^{(m-1})/2}{m^{mz-1/2}}\Gamma(mZ)$

.

Since
$\alpha_{\grave{J}}(_{n,\mathit{1}}\backslash ,$$= \frac{1}{\phi_{q}}\alpha_{\grave{J}}^{\prime_{n}}+A^{-\tau\backslash }’\frac{\dot{r}_{aA^{+\mathrm{J}}}-1}{\phi_{q}}$$(q=’.1,$ $.2\cdot,$$k^{\backslash }J$,

we have, by (5.2),$\prod_{J}\Gamma(\alpha_{J}^{(k)}+\frac{2s+iLv.+pLlp}{\phi_{0}\cdot\cdot\phi_{k}})$$= \prod_{J}^{-}(k1)j+1=\prod_{k1}^{k}\tau\phi(\frac{\alpha_{J}^{(k1)}-}{\phi_{k}}+\frac{j_{k+1}-1}{\phi_{k}}+\frac{2s+iLv_{p}+Llp}{\phi_{0}\cdots\phi_{k}}\mathrm{I}$$= \prod_{J}^{(1)}k-\Gamma(\alpha^{(}Jk-1)+\frac{2s+iLv_{\mathrm{P}^{+}}Llp}{\emptyset 0\cdots\emptyset k-1})$$\cross\prod_{J}^{)}\mathrm{t}(k-1(2\pi)^{(\emptyset)/2}k-1(\phi_{k})^{1}/2-\alpha-((k-1)2s+iLv+\mathrm{p}L1_{p})/(\emptyset 0\cdots\phi k-1)\}J$

Putting$(a-1)$$P_{a}= \prod_{J}(2\pi)^{(\emptyset-1)}a/2$$(a=1, \ldots, k)$$N_{a}= \prod^{)}(\emptyset(a-1Ja)-1+2\alpha_{J}^{(a}-1)$,$(a=1, \ldots, k)$$((1.7))$and using the induction and (1.4), we have$\prod_{J}\Gamma(^{(k)}\alpha+J\frac{2s+iLv.+pLlp}{\phi_{0}\cdot\cdot\phi_{k}})$$=P_{k}N_{k}^{-1}/2M_{k}-(2S+iLv_{\nu}+Ll_{p})$$\cross\prod_{J}^{-}\tau((k1)\alpha(k-1)J+\frac{2s+i.Lv_{p}+L\iota P}{\phi_{0}\cdot\cdot\phi_{k-1}})$$=P_{k}\cdots P_{1}(Nk\ldots N_{1})^{-}1/2(Mk\ldots M_{1})^{-}(2\theta+iLv+p1,L)$$\cross\prod^{(0}\tau(\alpha_{J}(j),+\frac{2s+iLv_{p}+L\iota_{p}}{\phi_{0}}0))$

.

287

MULTIPLJ SERIES IN A TOTALLY IMAGINARY NUMBER FIELD II$=(2\pi)^{\iota-}1/2(2\iota)1/2-(2s+iLv_{p}+L\mathrm{t}_{p})$$\cross P_{k}\cdots P_{1}(N_{k}\cdots N1)-1/2(Mk\ldots M1)-(2\mathit{8}+iLv_{p}+Ll_{\mathrm{p}})$$\cross\Gamma(2s+iLv_{p}+L\iota_{p})$,which gives (5.1) with$g(s, \lambda)$$= \prod^{t}.\{(2\pi)^{1/}2-1(2l)^{-}1/2+2s+iLvp+Llp$$p=1$$\cross$$(P_{k}\cdot.

.., $N_{1})^{1/2}(M_{k}$.P_{1})^{-1}(N_{k}$$=(2\pi)r/2-\iota\Gamma(2l)-r/2+2rS(P_{k}$...

.

., $M_{1})^{2\theta++L\iota_{p}}iLv_{p}\}$$P_{1})^{-r}(Nk$...

$N_{1})r/2(M_{k}\cdots M_{1})2rS$.

$\cross\prod_{p=1}’(2lM_{k}\cdots M_{1})^{L\iota_{p}}$Consequently we have
(4.3)

$H_{\lambda}(s, \tau..’\xi;\alpha J’ a_{J})$$=g(s, \lambda)(.2P\pi)^{-nS}$$\cross\prod_{p=1}^{r}\{\frac{(i\xi_{p})^{La_{P}}(i\xi_{p+}r)^{L}a_{p+r}}{(T_{p}^{2}+|\xi_{p}|2)(2S+iLv_{p}+L\iota_{\mathrm{p}})/2}\frac{F(2s+iLv_{p},L\iota,x)pp}{2^{2\theta+iL}v_{\mathrm{p}}+L\iota_{p}\tau(Ll_{p}+1)}.\}$$\cross\prod_{J}\{\Gamma(\frac{s}{P_{J}}+\alpha_{J}$;$\lambda^{L/P}J)\zeta(\frac{s}{P_{J}}+\alpha J’\lambda L/P_{J}$

According to Theorem 3.1, this expression (6.3) shows that, ifct ) is an entire function of
$J$

$s$

, and if$\lambda=1$, then$s=s_{J}=-0\alpha JP_{J}$Thus the proof is completed.

口$\emptyset_{J})\}$$\lambda\neq 1$$H_{1}(s, \tau,\xi;\alpha J’ aJ)$$(J\in J)$

and$s=SJ=(1-\alpha_{J})PJ$;$(J\in J)$.

.

, then$H_{\lambda}(S,$$T,$ $\xi;\alpha J$

has simple poles at

’

288

TAKAYOSHI MITSUI

Similarly to (5.1), we obtain$\prod_{J}\Gamma(1-\alpha_{J}-\frac{s}{P_{J}};\overline{\lambda}L/PJ)=g^{*}(s, \lambda)\prod_{1p=}^{r}\tau(1-2s-iLv_{p}+Ll_{p})$

,

where
(5.4)

$g^{*}(S, \lambda)$$= \prod^{t}\{(2\pi)^{\iota-1}/2(2\iota)^{-1/}2+2S+iLvp-Llp$$p=1$$\cross P_{k}\cdots P_{1}(N_{k}\cdots N1)^{1}/2(M_{k}$...

$M_{1})^{2s}+iLv_{\mathrm{p}}-L\iota_{p}\}$$=(2\pi)\iota r+r/2(2\iota)-r/2+2r\theta(P_{k1}\ldots P)^{r}(Nk\ldots N_{1})r/2(M_{k}\cdots M1)^{2s}r$$\cross\prod(2lM_{k}\cdots M_{1})^{-L\iota}r\mathrm{p}$

.

$p=1$

THEOREM 6.2.

$H_{\lambda}(S, \mathcal{T}, \xi;\alpha J’ a_{J})$satisfies the functional equation as follows:$H_{\lambda}(_{S,\mathcal{T}}, \xi;\alpha_{J}, a_{J})$$= \frac{(N_{1}\cdots N_{k})^{r}(M_{1}\cdots M_{k})^{r}}{D^{N/2}\prod JN(a_{J})}\prod_{p=1}^{r}\frac{1}{(\tau_{p}^{2}+|\xi_{\mathrm{P}}|^{2})1/2}\cdot H_{\overline{\lambda}}(1/2-s, \tau^{*}, \xi*;\alpha*a_{J}^{*}J’)$PROOF.
. (5.5)

By the functional equations (2.2) of zeta functions, we have$\lambda L/P$ $J$$\prod_{J}\{\Gamma(\frac{s}{P_{J}}+\alpha_{J;}\lambda L/PJ)\zeta(\frac{s}{P_{J}}+\alpha_{J},$; $aJ)\}$$= \prod_{J}\frac{(2\pi)^{r\mathrm{t}^{2}(S}\alpha_{J}+/PJ)-1\}}{N(a_{J})\sqrt{D}}$$\cross\prod_{J}\{\Gamma(1-\alpha_{J}-\frac{s}{P_{J}}$;$\overline{\lambda}^{L}/P_{J}$$\overline{\lambda}L/P_{J})((1-\alpha j-\frac{\mathrm{s}}{P_{J}}.$’We shall show that
(5.6)$\sum_{J}\alpha_{J}=\frac{1}{2}(N-1)$,; $aJ)*\}$ .

.

289

MULTIPLE SERI

$J\mathrm{S}$IN A TOTALLY IMAGINARY NUMBER FIII$J\mathrm{L}\mathrm{D}$where $N= \sum_{J}1$ . In fact,.

$\sum_{J}\alpha_{J}=\sum_{Ja}\sum_{=0}\frac{j_{a+1}.-1}{\phi_{a}\cdot\cdot\phi_{k}}k=\sum_{0a=}^{k}\sum_{J}\frac{j_{a+1}.-1}{\phi_{a}\cdot\cdot\phi_{k}}$$= \sum_{0a=}^{k}\sum_{J}^{a)}\frac{j_{a+1}-1}{\phi_{a}}(=\sum_{=j_{1}1}^{\phi}0.\frac{j_{1}-1}{\phi_{0}}+a\sum_{1=}^{k}\sum_{J}^{(a-1)}\frac{\phi_{a}-1}{2}$$= \frac{1}{2}(\emptyset 0-1)+\frac{1}{2}a\sum(=1k\sum_{J}^{()}a1-\sum 1(a-1)J)=\frac{1}{2}(N-1)$.

Further we have

.

(5.7)

$\sum_{J}\frac{1}{P_{J}}=\sum_{J}\frac{2}{\phi_{0}\cdots\phi_{k}}=2$

By (5.6) and (5.7)(5.8)$\prod_{J}\frac{(2\pi)^{r;2}(\alpha_{J}+s/PJ)-1\}}{N(a_{J})\sqrt{D}}=\frac{(2\pi)^{4r}S-r}{D^{N/2}\prod_{J}N(a_{j})}$.

By (5.8), (5.4) and (5.5),(5.9)$\prod_{J}\{\Gamma(\frac{s}{P_{J}}+\alpha_{J};\lambda^{L}/P_{j})\zeta(\frac{s}{P_{J}}+\alpha_{J},$$\lambda^{L}/P;Ja_{J})\}$$= \frac{(2\pi)^{4s-}rr}{D^{N/}2\prod_{J}N(a_{J})}g^{*}(S_{)}\lambda)$$\cross\prod_{p=1}\Gamma(1-2s-iLv_{p}+Ll_{p})r\prod_{J}\zeta(1-\frac{s}{P_{J}}-\alpha J’\overline{\lambda}^{L}/P;Ja_{J)}^{*}$

Putting (5.9) into (5.3),$H_{\lambda}(s,\tau,\xi;\alpha J’\alpha_{J})$$= \frac{(2\pi)^{4}rs-r}{D^{N/2}\prod_{J}N(a_{J})}g(S, \lambda)g^{*}(S, \lambda)(2P\pi)^{-nS}$.

$\cross\prod_{p=1}^{r}\{\frac{(i\xi_{p})La_{p}(i\xi \mathrm{P}+r)La+\mathrm{p}\gamma}{(\tau_{p}^{2}+|\xi_{\mathrm{P}}|2)(2s+*LvP^{+}p)Ll/2}$$\cross\frac{\Gamma(1-2_{S}-iLvp+Ll_{p})}{2^{2s+iL}v+p\mathrm{t}_{p}L\Gamma(Ll_{p}+1)}F(2s+iLv_{p}, Ll_{p} ; X_{\mathrm{P}})\}$

.

290

TAKAYOSHI MITSUI$\cross\prod_{J}\zeta(1-\frac{s}{P_{J}}-\alpha_{J},\overline{\lambda}^{L}/P_{J}.$;$a_{J}^{*})$$= \frac{(N_{1}\cdots N_{k})^{r}}{D^{N/2}\prod JN(a_{J})}P^{r}2^{2nS}-rl-r(2P\pi)^{n\theta}-r$$\cross$$\prod_{-,p=\mathrm{i}}^{r}\dagger_{\backslash }\frac{(i\xi_{p})^{La}p(i\xi_{p+}r)La-_{p+\prime}}{(_{\mathcal{T}_{\mathrm{P}}^{2}}+|\backslash \xi_{s}\vee \mathrm{r}|^{2}\text{ノ})(2s+iLv+pl_{p}L)/2}$$\cross\frac{\Gamma(1-2s-iLv+p)L\iota_{p}}{2^{2S+iL}v+pL\iota\Gamma p(L\iota_{p}+1)}F(2s+iLv_{\mathrm{P}}, L\iota_{P} ; X)p\}$$\mathrm{x}\prod_{J}((1-\frac{s}{P_{J}}-\alpha_{J},\overline{\lambda}L/P_{J}$Let$\tau_{1}^{*},$$\tau_{r}*$$\ldots,$and$\xi_{1}^{*},$$\xi_{r}^{*}$$\ldots,$;$a_{J}^{*)}$.

be the numbers defined by (1.8), then we easily see that$(p=1, \ldots, r)$$\tau_{p}=\frac{\tau_{p}^{*}}{\tau^{*2}+|p\xi^{*}p|2}$$\xi_{p}=\frac{\xi_{p+r}^{*}}{\tau_{p^{2}p}^{*}+|\xi^{*}|2}$,,$\xi_{p+r}=\frac{\xi_{p}^{*}}{\tau_{p}^{*^{2}}+|\xi_{p}^{*}|2}$$\tau_{p}^{*2}+|\xi^{*}p|2\frac{1}{\tau_{p}^{2}+|\xi_{p}|^{2}}=$$(p=1, \ldots, r)$$(p=1, \ldots,\Gamma)$$x_{p}= \frac{|\xi_{p}|^{2}}{\tau_{p}^{2}+|\xi_{p}|^{2}}=\frac{|\xi_{p}^{*}|^{2}}{\tau^{*2}+|p\xi p|^{2}*}=x_{p}^{*}$,,$(p=1, \ldots, r)$,and(5.10)Let$\overline{(\tau}_{p}^{\frac{(i\xi_{p})^{La}p(i\xi p+r)La_{p+\Gamma}}{2+|\xi_{p}|^{2})(2S+iLvp+L\iota_{p})/2}=}\frac{(i\xi_{p+r}^{*})La_{\mathrm{p}}(i\xi p)^{L}*a_{p+r}}{(\tau_{\mathrm{P}}^{*^{2}}+|\xi^{*}p|^{2})(-2S-iLv\mathrm{p}+Ll_{p})/2}$$\alpha_{J}^{*}(J\in J)$(5.11)

be the numbers defined by (1.3), then
$1- \frac{s}{P_{J}}-\alpha_{J}=\frac{1}{P_{J}}(\frac{1}{2}-s)+\alpha_{J}^{*}$

Using (5.10), (5.11) and (1.5), we write$(5.!2)$$H_{\lambda}(_{S,\tau}, \xi|.\alpha_{J}, a_{J})$ns-r ご (2P\mbox{\boldmath $\pi$})ns-r$= \frac{(N_{1}\cdots N_{k})^{r}}{D^{N/2}\prod JN(a)j}P^{r}2^{2}$$(J\in J)$$(p=1, \ldots, r)$

.

291

$\mathrm{M}\mathrm{U}\mathrm{L}\dot{\mathrm{T}}$IPLE SERIES IN A TOTALLY IMAGINARY NUMBER FIJLD II.
$\cross\prod_{p=1}^{r}\{\backslash \frac{(i\xi_{p+r}*)La(pi\xi_{p}^{*})^{L}a_{p+}}{(_{\mathcal{T}_{p}^{*^{2}}}+|\xi^{*}p|2)(-2S-iLv+PL\iota_{p})/2},$$\cross\frac{\Gamma(1-2s-iLvP^{+}L\iota_{P})}{\mathit{2}^{2\theta+}iLv_{p}+L\iota_{p}\Gamma(Ll_{p}+1)}F(1-2s-iLv_{p}, Ll;x_{p}^{*})p\}$$\cross\prod_{J}\zeta(\frac{1/\mathit{2}-s}{P_{J}}+\alpha_{J}^{*},\overline{\lambda}L/P_{J};.a_{j)}^{*}$$= \frac{(N_{1}\cdots N_{k})^{r}(M_{1k}M)r}{D^{N/2}\prod JN(\alpha_{J})}\ldots\prod_{p=1}^{r}\frac{1}{(\tau_{p}^{2}+|\xi_{p}|^{2})^{1}/2}$$\mathrm{x}(\mathit{2}.P\pi)^{-}n(1/2-s).\prod_{p=1}^{r}\{\frac{(i\xi_{p+r}^{*})^{La}p(i\xi_{p}*)^{La}p+r}{(\tau_{p}^{*2}+|\xi^{*}p|2)(1-2\theta-iLvp+L\iota p)/2}$$\cross\frac{\Gamma(1-\mathit{2}s-iLv+Lpp\iota)}{2^{1-2iL}S-v+pL\iota p\Gamma(Ll_{p}+1)}F(1-\mathit{2}s-iLv_{p}, L\iota_{p} ; X_{P}^{*})\}$$\mathrm{x}\prod_{J}((\frac{1/2-s}{P_{J}}+\alpha_{J}^{*},\overline{\lambda}L/Pj$;.

$a_{J}*)$

Comparing the last expression of (5.12) with (4.18), we have
(5.13)$H_{\lambda}(s, \tau, \xi;\alpha J’ a_{J})$$= \frac{(N_{1}\cdots N_{k})^{r}(M_{1k}M)^{r}}{D^{N/2}\prod JN(a_{J})}\ldots\prod_{p=1}^{r}\frac{1}{(\tau_{p}^{2}+|\xi_{p}|^{2})^{1}/2}$$\cross H_{\overline{\lambda}}(1/\mathit{2}-s, \tau^{*}, \xi^{*} ; \alpha_{J}^{*})a_{J})*$.

口

Let $N_{a}^{*}(a=1, \ldots, k)$ be the numbers defined by (1.8), then
(5.14)$N_{a}N_{a}^{*}M^{2}=a1$$(a=1, \ldots, k)$.

In fact,$N_{a}N_{a}^{*}M_{a}^{2}= \prod^{)}(\emptyset a)^{-}(a-1J1+2\alpha_{J}-1)-1(l+2\alpha_{J}^{\prime(-}+\iota 1)2/\phi 0\cdots\psi a-1$

292

TAKAYOSHI MITSUI

and$-1+2 \alpha_{J}^{(a}-1)-1+2\alpha_{J}*(a-1)+\frac{2}{\phi_{0}\cdots\phi a-1}$..

$+ \frac{\phi_{a-1^{-}}j_{a}}{\phi_{a-1}})$$=-2+2( \frac{j_{1}.-1}{\phi 0^{\cdot}\cdot\phi_{a-}1}+\cdots+\frac{j_{a}-1}{\phi_{a-1}})+2(\frac{\phi_{0^{-j_{1}}}}{\phi 0\cdots\phi a-1}+\cdot$$+ \frac{A}{\phi 0^{\cdot}\cdot\phi a-1}‘.=0$,

which shows that
$N_{a}N_{a}^{*}M^{2}=a1$.

We now put
If(5.15)$( \mathcal{T}, \xi;\alpha_{J}, aJ)=\frac{D^{N/2}\prod_{J}N(.a_{J})}{(N_{1}\cdots N_{k})^{r}(M_{1}\cdot\cdot Mk)^{r}}\prod_{p=1}r(\mathcal{T}^{2}p+|\xi p|2)1/2$

and(5.16)$K( \tau^{*}, \xi^{**} ; \alpha_{J}, a^{*}j)=\ldots\frac{D^{N/2}\prod_{J}N(a_{J})*}{(N_{1k}^{*}N^{*})^{r}(M_{1}\cdots Mk)^{r}}\prod_{p=1}(\tau+*2|p\xi^{*}p|^{2})r1/2$

Then (5.14) gives that
(5.17)$I\iota’(\tau, \xi;\alpha J)aJ)K(T\xi^{*} ; \alpha J’ aj)=1*,**$and (5.13) is written as follows:(5.18)$H_{\lambda}(S, \tau, \xi;\alpha_{J}, \alpha_{J})=K(\tau\xi^{**}*,.)

THEOREM 63(5. I9)In the strip; \alpha J’ a^{*}JH\ddot{\lambda}(1/2-s, \tau\xi^{*}*,J ; \alpha^{*}, a_{J})*$$|\sigma|\leq 1+1\mathrm{n}\mathrm{a}\mathrm{x}_{J}\epsilon \mathcal{J}\mathrm{t}P_{J}$}, we ave

$H_{\lambda}(_{S,\tau}, \xi;\alpha_{J}a_{J}))\ll e-c|t|$where $c>0$ and the constants in his est imation depend
$t$

$h$

,
on

$\tau,\xi,$$\alpha_{J},$$a_{J}$

and$\lambda$.

293

MULTIPLE SERIES IN A TOTALLY IMAGINARY NUMBER FIELD IIPROOF.

According to (5.18), it is sufficient to prove Theorem under the condition$0 \leq\sigma\leq 1+\max_{J\in J}\{P_{J}\}$. Rom (4.18) we have

$H_{\lambda}(s, \tau, \xi;\alpha J, a_{J})$$\ll|_{p=1}\prod^{r}G(2S+iLv_{p}, Ll;ppX)\prod\zeta J(\frac{s}{P_{J}}+\alpha_{J},$$\lambda^{L/P}\cdot;$; $aJ)|$ .

Theorem 42 and Theorem 32 give$G(2s+iLv_{p}, L\iota_{p} ; x_{p})\ll(1+Ll_{p}+|2t+Lv_{p}|)^{2\sigma-}1/2e-c|2t+Lv_{p}$$\ll e^{-c|t|}$

I

.

and
$\zeta(\frac{s}{P_{J}}+\alpha_{J},$$\lambda^{L}/P_{J}$; $a_{J})\ll(1+|t|)^{2n}$ ,respectively. Hence (5.19) is obtained at once.

$\square$\S 7. Functional equation and residues
Theoem 63shows that$\int_{\sigma_{1}+}^{\sigma_{0}+}i\tau\lambda H(_{S},\tau iT,\xi;\alpha_{J}, aj)dsarrow 0$$(|T|arrow\infty)$

,

where
$\sigma_{1}=-\max\{(1-J\alpha*J)P_{J}\}-\frac{1}{2}$and the integral is taken along the horizontal line from
Hence, in the expression of

integraion to the vertical line
If
(6.1)

$\lambda\neq 1$$T(\tau, \xi;\alpha_{J}, a_{J})$$\sigma=\sigma_{1}$.

$\sigma_{1}+iT$to$\sigma_{0}+iT$.

obtained in (4.17), we can shift the path of.

, then$\frac{1}{2\pi i}\int_{(\sigma_{0})}H_{\lambda}(s, \tau,\xi;\alpha J’\emptyset_{j})ds=\frac{1}{2\pi i}\int_{(\sigma_{1}})H_{\lambda(S},T,$ $\xi;\alpha

J’ aJ)ds$.

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TAKAYOSHI MITSUI

By (5.18),(6.2)$\frac{1}{2\pi i}\int_{(\sigma_{1})}H\lambda(s, \mathcal{T}, \xi;\alpha J’ aJ)d_{S}$$=K(_{\mathcal{T}^{*}}, \xi^{*} ; \alpha^{**}J’ a_{J})\frac{1}{2\pi i}\int_{(\sigma_{1})}H_{\overline{\lambda}}(1/2-S, \tau^{*}, \xi^{*} ; \alpha^{**}J’ aJ)d_{S}$1$=I^{-}\acute{\mathrm{C}}(-.\tau^{:}..,\dot{\zeta}.

$[$$\prime\prime\sigma’\cdot-\alpha^{\mathrm{r}*}*,’*,J’\check{\mathrm{u}}J^{\backslash }Jd_{\delta}$; \alpha_{J}^{\sim}, a_{\dot{J}}.\grave{J}]_{(\sigma_{0}^{*}}\overline{2\pi i})\sim H\overline{\lambda}(6,$

,

where
$\sigma_{0}^{*}=\max_{J}\{(1-\alpha_{J}*)PJ\}+1$

If$\lambda=1$.

, we have similarly(6.3)$\frac{1}{2\pi i}\int_{(\sigma 0)}H_{1}(\mathit{8}, T, \xi;\alpha J’ aJ)d_{S}$$= \frac{1}{\mathit{2}\pi i}\int_{(\sigma_{1})}H_{1}(_{S,\mathcal{T}}, \xi;\alpha J’ a_{j})ds+R(\mathcal{T}, \xi;\alpha J’ a_{J})$$=I \mathrm{t}^{r}(\mathcal{T}^{*}, \xi^{***}).J’)\alpha a_{J}\frac{1}{2\pi i}\int_{(\sigma_{0})}.H_{1}(S, \tau^{*}, \xi^{*} ; \alpha J’ a_{J}^{*})*dS+R(\tau, \xi;\alpha_{J}, a_{J})$

,

where
(6.4)

$R( \tau, \xi;\alpha J’ aJ)=\sum_{\sigma\sigma\iota\leq\leq\sigma 0}{\rm Res}_{\sigma}H1(_{S}, \tau, \xi;\alpha_{J}, a_{J})$.

Hence by (6.1), (6.2) and (6.3),(65)$T(\tau, \xi;\alpha j’ a_{J})$$=A \sum_{\lambda}\frac{1}{2\pi i}\int_{(\sigma_{0})}H_{\lambda}(s,\tau, \xi;\alpha J’\alpha J)d_{S}$$=AK( \tau^{*}, \xi^{**}|.\alpha J’ aj)*\sum_{\lambda}\frac{1}{2\pi i}\int_{(\sigma_{0})}.H_{\lambda}(s,\mathcal{T}\xi^{*}*,*)i\alpha Ja^{*}J)ds$$+AR(\tau,\xi;\alpha J’ aJ)$$=K(\tau^{*},\xi**;\alpha_{J}, a^{*}J)\tau(\tau^{**},\xi ; \alpha^{*}J’ a^{*}J)+AR(\tau,\xi;\alpha J’ a_{J})$.

We divide the sum (6.4) into two parts:(6.6)$R( \tau,\xi;\alpha J’ aJ)=\sum_{J}{\rm Res}_{S=s_{J}}H1(S, \mathcal{T},\xi;\alpha_{J’ J}a)+\sum_{J}{\rm Res}_{=}H1(sSs_{J}^{0}’ \mathcal{T},\xi;\alpha_{J},\alpha J)$

.

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MULTIPLE SJRIES IN A TOTALLY IMAGINARY NUMBER FIELD II

Here we see(6.7)${\rm Res}_{s=s_{j}^{0}}H_{1}(_{S}, \mathcal{T}, \xi;\alpha J’ aJ)$$= \lim_{sarrow sJ}(s-0s_{J})0H_{1}(S, \tau, \xi;\alpha_{J}, a_{J})$$=I_{\acute{1}}(_{\mathcal{T}^{*}}, \xi*;\alpha_{J}, a_{J}^{*}*)sarrow_{S}\lim(0JS-s_{j})0H_{1}(1/\mathit{2}-S, \mathcal{T}\xi*,*;\alpha_{J’ J}a**)$.

If we put
$S_{J}^{*}=(1-\alpha^{*}J)PJ$$((1.10))$ ,then$\frac{1}{2}-s_{JJ}^{0}=s^{*}$and so (6.7) is(6.8)${\rm Res}_{s=s_{J}^{0}}H_{1}(_{S,\mathcal{T}}, \xi;\alpha_{J’ J}\alpha)$$=-I \acute{\backslash }(\tau^{*}, \xi^{**} ; \alpha J’ a^{*}J)\lim(_{S}sarrow s^{*}J-s_{J})*H_{1}(S, \tau^{*}, \xi*;\alpha_{j}*,*aJ)$$=-I\mathrm{f}(\mathcal{T}^{****}, \xi|.\alpha J’ a^{*}J){\rm Res}_{\theta=S_{J}^{*}}H1(S, \mathcal{T}, \xi^{*} ; \alpha^{*}J’ a_{J}^{*})$.

Thus we have by (6.6) and (6.8)(6.9)$R( \tau, \xi;\alpha J’ aJ)=\sum{\rm Res}_{s=s_{J}}H_{1}(_{S}, \tau,\xi;\alpha J’ aJ)J$$-I \acute{\mathrm{L}}(_{\mathcal{T}^{*}}, \xi*\alpha_{J};a_{J}*,*)\sum_{J}.\cdot{\rm Res}_{*,=sJ}H1(s, \tau^{*}, \xi*;\alpha_{J}SJ*,*a)$.

Rom the results (6.5) and (6.10), we have
(6.10)$T(\tau, \xi;\alpha J’ aJ)=I\zeta(_{\mathcal{T}^{*}}, \xi^{*} ; \alpha^{***}J’ aJ)\tau(_{\mathcal{T}}, \xi^{**} ; \alpha J’ aJ)*$$+A \sum_{J}s=’;{\rm Res}_{s_{J}}H_{1}(s,\mathcal{T}\xi\alpha J’ aJ)$$-AK( \tau^{*}, \xi^{*} ; \alpha^{*}j’ aj)*\sum_{j}={\rm Res}_{Ssj}H_{1}(S,\tau^{*},\xi^{**} ; \alpha_{J}, a_{J}^{*})$

.

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TAKAYOSHI MITSUI

Further by (5.17), we write (6.10) in a symmetric form:(6.11)$I \mathrm{s}’(.T,.\xi;\alpha J’ aJ)^{1/}2\{T(\mathcal{T}, \xi;\alpha J’ aJ)-A\sum_{J}S=S({\rm Res} H1S, \mathcal{T}, \xi;\alpha J’ aJ)\}J$$=I\iota^{r}(_{\mathcal{T}}.*, \xi*;\alpha.\Gamma, a_{r}.)^{1}**/2\{T(\tau^{**}, \xi ; \alpha^{**}J’ a_{J})-A{\rm Res}_{\frac{7^{\wedge}}{J}J}H_{1}(s, \tau\xi*,**);\alpha J’ a_{J}^{*}\}-*J^{\cdot}$$s-s_{J}$Putting$\frac{s}{P_{J}}+\alpha_{J}=1+\frac{s-s_{J}}{P_{J}}$,we have${\rm Res}_{s=_{S_{J}}} \zeta(1+\frac{s-s_{J}}{P_{J}},$$a_{J)}= \frac{P_{J}(2\pi)^{\Gamma}R}{wN(a_{J})\sqrt{D}}$.

Hence we obtain from (4.19)${\rm Res} H_{1}(_{S,\mathcal{T}}, \xi;\alpha_{J’ J}a)$$s=s_{J}$$= \frac{(2\pi)^{r}R}{w\sqrt{D}}\frac{P_{J}\Gamma(2SJ)^{r}}{N(a_{J})(\mathit{2}^{2}P\pi)ns_{j}}$$1$$\cross\prod_{p=1}^{r}\frac{1}{(\tau_{p}^{2}+|\xi p|2)^{\theta}j}F(SJ’\frac{1}{2}-S_{J},$;$\frac{|\xi_{p}|^{2}}{\tau_{p}^{2}+|\xi_{p}|^{2}})$$\cross(J_{1}^{1}\neq J\in\prod_{)J}\zeta(1+j\frac{s_{J}-s_{J_{1}}}{P_{J_{1}}}, \alpha J\iota)\cdot$

and by the definition (1.11) of(6.12)$S(\tau, \xi;\alpha_{J}, a_{J})$, we have$S( \tau, \xi;\alpha_{J’ J}\alpha)=A\sum\theta=\theta)J{\rm Res} Hr1(s,\tau,\xi;\alpha J’ a_{J}$Sinilarly(6.13)$S( \tau^{*}, \xi^{*} ; \alpha J’ aJ)**A=\sum Jss_{J}{\rm Res}_{=}H_{1}\mathrm{r}(S, T^{*},\xi^{*} ; \alpha J’ a**J)$.

Collecting the results (6.11), (6.12) and (6.13) and noting (5.15) and (5.16), we finall

obtain Main Theorem.

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$J\mathrm{S}$IN A TOTALLY IMAGINARY NUMBJR FIJLD II

References[1] J. HECKE, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen II,
Math. Z. 6 (1920), 11-51.
[2] T. MITSUI, On the partition problems in an algebraic number field, Tokyo J. Math. 1 (1978), 189-235.
[3] T. MITSUI, On certain multiple series with functional equation in a totally imaginary number field I,
Tokyo J. Math. 18 (1995), 61-74.
[4] H. RADEMACHBR, Zur additiven Primzahltheorie algebraischer Zahlk\"orper III, Math. Z. 27 (1928),

321-426.

Present Address
DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, GAKUSHUIN UNIVERSITY,
MEJIRO, TOSHIMA-KU, TOKYO, 171 JAPAN