$B9)A!Bg$NDMK\$H?=$7$^$9(B.

In article <d43e5a16-c3ca-4372-9166-9aecd813dc22@g26g2000vbi.googlegroups.com>
KyokoYoshida <kyokoyoshida123@gmail.com> writes:
> http://beauty.geocities.jp/yuka26076/study/Number_Theory/def566.JPG
> http://beauty.geocities.jp/yuka26076/study/Number_Theory/theorem1_2.JPG
> (i)$B&F(B(s)$B$OJ#AGA4J?LL$KM-M}7A4X?t$H$7$F2r@O@\B3$G$-(B,s=1$B$r=|$$$F@5B'$G$"$k!#(B
> (ii) s=1$B$O&F(B(s)$B$N(B1$B0L$N6K$G$"$j(B,$B$=$3$G$NN1?t$O(B1$B$G$"$k!#(B
> (iii) $BG$0U$N(Bs$B$KBP$7(B,$BEy<0&P(B^{-s/2}$B&#(B(s/2)$B&F(B(s)=$B&P(B^{-(1-s)/2}$B&#(B((1-s)/2)$B&F(B(1-s)$B$,(B
> $B@.$jN)$D(B($BC"$7(B,$B&#$O(Bgamma$B4X?t(B)$B!#(B
> 
> $B$NLdBj$r>ZL@$r;n$_$F$*$j$^$9!#(B
> 
> (i)$B$N>ZL@$K4X$7$F(B
> $B@h$:&F4X?t$O(Bs=1$B$G$ODj5A$5$l$F$J$$$N$G(B
> "s=1$B$r=|$$$F(B"$B$N2U=j$,0UL#ITL@$J$N$G$9$,$3$l$O$I$&$$$&$3$H$G$7$g$&$+(B?

$B$=$3$G$ODj5A$5$l$F$$$J$$$J$i(B, s = 1 $B$r=|$/$N$OEvA3$G$7$g$&(B.
$B$^$"(B s = 1 $B$O6K$K$J$k$N$G(B, $BCM$H$7$FL58BBg$r(BC$B";(Ba$B$H$7(B,2$B=EO"7k(BD$B":(B
> {C_1$B".(BC_2$B":(B2^B;C_1:={z$B":(BC;|z-a|$B!e(Br_1} and C_2:={z$B":(BC;|z-a|<r_2},
>               r_1,r_2$Bch$(D??(B, r_1>r_2}
> http://beauty.geocities.jp/yuka26076/study/Number_Theory/img003.jpg
> $B$K$F&F4X?t$OL@$i$+$K(BD$B>e$G@5B'$J$N$G(B
> ($B"h&F4X?t$NDj5A$h$j&2(B_{n=1}^$B!g(B1/n^s$B$O"O(Bs$B":(BD$B$K$FHyJ,2DG=(B
> ($B"h&F4X?t$O(BD$B>e$G0lMM<}B+$9$k$N$G(B($B"h(B??)))$BJ,$+$k!#(B

$B$=$s$J$3$H$G$OJ,$+$j$^$;$s(B.

> $B$3$3$N(B??$B$NM}M3$,J,$+$j$^$;$s!#$I$&$9$l$P<($;$^$9$G$7$g$&$+(B?

$BLdBj$O$=$l$h$jA0$K$"$j$^$9(B.

> $B$=$7$F(B,$Be$G&F$,2r@O@\B3$G$"$k;v$r<($7$?$$$N$G$9$,(B
> $B2r@O@\B3$NDj5A$O(B
> http://beauty.geocities.jp/yuka26076/study/Number_Theory/def566.JPG
> $B$N(BDef572$B$NDL$j$@$H;W$$$^$9!#(B
> $B:#4X?t$O&F$@$1$G(Bg$B$KAjEv$9$k$b$N$,L5$/$F:$$C$F$$$^$9!#(B
> $B$I$&$d$C$F2r@O@\B3$G$"$k;v$,<($;$^$9$G$7$g$&$+(B?

$BDL>o(B, \Gamma(s) \zeta(s) $B$N@QJ,I=<($rMQ$$$^$9(B.

> (ii)$B$N>ZL@$K4X$7$F(B
> $B&F$r(BLaurent$B$NDjM}$r;H$C$FNN0h(BD$B$G(BLaurent$BE83+$9$k$H&F(B(s)=$B&2(B_{k=1}^$B!g(B1/n^s
> =$B&2(B_{k=0}^$B!g(Bc_k(s-a)^k+$B&2(B_{k=1}^$B!g(Bb_k/(s-a)^k ($BC"$7(B,b_k,c_k$B":(BC)$B$H$J$k(B
> ($BC"$7(B,c_k=1/(2$B&P(Bi)$B"i(B_c$B&F(B(s)/(s-a)^{k+1}ds,
> b_k=1/(2$B&P(Bi)$B"i(B_c$B&F(B(s)(s-1)^{k-1}ds
> $B$G@QJ,7PO)(Bc$B$O$$$:$l$bH?;~7W2s$j$H$9$k(B)$B!#(B
> 1$B0L$N6K$G$"$k;v$r8@$&$N$@$+$i(B
> $B$3$l$+$i(Bb_1$B!b(B0$B3n$D(Bb_2=b_3=$B!D(B=0$B$G$"$k;v$,8@$($l$P$h$$!#(B
> b_1=1/(2$B&P(Bi)$B"i(B_c$B&F(B(s)(s-1)^{1-1}ds=1/(2$B&P(Bi)$B"i(B_c$B&F(B(s)(s-1)^0ds
> =1/(2$B&P(Bi)$B"i(B_c$B&F(B(s)1ds=1/(2$B&P(Bi)$B"i(B_c$B&F(B(s)ds$B!b(B0
> (\xE2^H$B5?(B?)
> $B$3$3$O(BCauchy$B$N@QJ,8x<0(B
> $B!XJ#AG4X?t(Bf(z)$B$,C1=cJD6J@~(Bc$B$H$=$NFbIt(BD$B$G@5B'$G$"$k;~(B,
> D$BFb$NG$0U$NE@&A$K$D$$$F$A$K8@$($^$9$M!#(B

$B$$$d(B, b_1 = 1 $B$r<($9$N$G$9$h(B. $B$H$b$"$l(B,
 \Gamma(s) \zeta(s) $B$K$D$$$F$N@QJ,8x<0$rD4$Y$J$$$H(B,
$B2?$b=P$F$-$^$;$s(B.

> (iii)$B$K4X$7$F(B
> ($B:8JU(B)=$B&P(B^{-s/2}$B"i(B_[0..$B!g(B]exp(-x)x^{s/2}dx$B&F(B(s) ($B"h(Bgamma$B4X?t$NDj5A(B)
> ($B1&JU(B)=$B&P(B^{-(1-s)/2}$B"i(B_[0..$B!g(B]exp(-x)x^{(1-s)/2}dx$B&F(B(1-s)
> $B$+$i0lBN$I$&$9$l$P$$$$$N$G$7$g$&$+(B

$B$3$l$bJL$N@QJ,I=<(<0$+$iF@$i$l$k$N$G$9$,(B,
$B$H$b$"$l(B, $B9T$-Ev$?$j$P$C$?$j$K9M$($F$b2r7h$7$^$;$s$+$i(B,
$B@lLg=q$KEv$?$C$FJY6/$5$l$k$3$H$r$*4+$a$7$^$9(B.

$B:#(B, $BO$@$1FI$a$PNI$$H&$G$9(B.
 (iii) $B$O?tO@#2$NBh#7>O$N=i$a$NJ}$K$"$kDjM}(B7.1$B$G$9(B.
$B$=$3$@$1FI$s$G(B, $BNI$/9M$($l$P(B, $BJ,$+$j$^$9(B.
-- 
$BDMK\@i=)(B@$B?tM}!&<+A3ItLg(B.$B4pHW2J3X7O(B.$B5~ET9)7]A!0]Bg3X(B
Tsukamoto, C. : chiaki@kit.ac.jp