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ñ ×صùœÔ-Ö-ãYØ%è°Ö-Ü-×ÝsöÝ
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þ è¥Ý›Øé§Ùªäý[ìÕ§ÞØÕ§ãîÔ-Ö-ìÙØ (0, . . . , 0) êéäÖ¥ëªà
|x1 . . . xn | = c,
cø Ýsö›å2ßÐæÔsÝ
Ù#Ö-æâéäÖ-ægÔ-Ö-ìÕ§ÙÚ×՜ó՜Ô-ÙØ
ÞØÕ§ã
c = 1, 2, 3, . . .
M
õä՜öÝsûÕ§æ
Ö-ñØé§ßÐêÝ
՜Ô-é£ëÚé§Ö-Ö
Ô-ê-՜Ô-éœÔ-ê
Ûªý_ûØãYÞÖ-×ãlÕ§ÞÞßÐã Ûª×Ý
ê-ÞÕð
M
|NK (µ)| = c,
n©
òºå2Õ
oqpGoqpsrt3uNvwx
µ∈D
÷-¸Âè°ê-Õ§éœÔ-ÞÖõì%Ô-Öçê-é§ÕcÕ§ò§Ög×՜óÕ§ÞØ%ëâå2Ö-ñ%Û-é§ÙªÝ
ý_ÔÚÖ§å2ÞÖ
è°ÞÝ¥ìÞÖ-Õcñ×Õ¹å2éœÔsÝ
êsäÕ§ÞØÕêê-Ø#å2Õ
µ = ±εk11 . . . εkuu µj ,
k = (k1 , . . . , ku ) ∈ Zu ,
ºò å2Õ {ε , . . . , ε } EÞÕ§Ù#Ö
Ô-Ö-×Ýsë†ácØÙ#é§Ø×Ö-êÝ
ÞÞÝsëxé§ØéœÔ-Õ§ãÝúÖ-é§ÞÖ-ê-ÞßàÿÕ¹å2ØÞØíxÙ#Ösäô-íÝ
u
Ù#Ö
Ô-Ö-×ß1 à
D
õìØéäÖ
u = n − 1,
Ý
CÞÕ§Ù#Ö
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ÞÞßÐÕÚ×՜óÕ§ÞØ%ë™õ
÷ .Ô-ÖÎÛ#Ô-ê-Õ§×ö›å2Õð
j (c) ∈ D
ÞصÕj ä=Õ§ò§Ù#µÖ¡
ê-ßÇÔ-Õ§ÙªÝ
՜ÔúØ%èÚÙªä2Ý
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Ûª. .×.ÕÒ, Q(c)
ò§×%Ûªññß Õ¹å2ØÞØí þ Ô|÷‹Õ
÷
Ö-Ü-×ÝsÔ-ØãlßàµùäÕ§ãlÕ§Þ%Ô-Ö-lê ÐÙ#Ösäô-íÝíÕäßàµìØé§ÕäÎñ×Ö-Ø%è°ê-Ösäô-ÞÖ-ò§ÖgñÖsäëÝ|äò§Õ§Ü-×Ý
ØìÕ§é§ÙتàµìØé§Õä*÷
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zaäëÚتääý[éœÔ-×Ý
íØØÒé‹àÕ§ãlß(×Ý
é§é§ãlÖ
Ô-×Øã†Ö§å2Øޙõ%ê!íÕäÖ-ãÿÔ-ØñØìÞßÐæ™õñ×Øãlէי÷NÂÛ-éœÔ-ôÞÝ
é[ØÞ%Ô-Õ§×Õð
éœÛ-՜ÔâÝ
é§Øãlñ%Ô-Ö
Ô-ØÙªÝgéœÛªãlãlß
X
S(h, Mφ , s1 , s2 ) =
ñ×Ø
h→0
ݛէéäØ
÷-ïÖ-ÙªÝsö›Õ§ãÐõì%Ô-Ögñ×Ø
s2 = s 1 > 1
1
(0,0)6=(x,y)∈Mφ
s 2 > s1 > 1
(1 +
|h−1 x|s1 )(1
é§ñ×Ý
ê-Õ¹åäØêÝÖ-íÕ§ÞÙªÝ
+ |h−1 y|s2 )
S(h, Mφ , s1 , s2 ) h2s1 ,
õ%Ô-Ö
S(h, Mφ , s1 , s2 ) = C ∗ h2s1 (ln h−1 + O(1)),
òºå2Õ
C∗ = 4
þ ©l
5s1 /2
ζK (s1 ).
ln ε
zÖ-ÙªÝsè¥ÝsÔ-Õäô-éœÔ-ê-ÖçùœÔ-Ö-ò§ÖgÛ#Ô-ê-Õ§×ö›å2Õ§ÞØ%ë×Ýsè°Ü-ØêÝ
՜Ô-é£ëÞݛÞÕ§é§Ù#Ösäô-Ù#ÖçùœÔsÝ
ñÖ-ê÷
È #$~漃 ~lË '&”Ì͏ï
¸ ãlÕ§Õ§ã
òºå2Õ
Q(c)
∞ X
X
X
1
S(h, Mφ , s1 , s2 ) =
=
2
F (k),
−1 µ cos φ|s1 )(1 + |h−1 µ0 sin φ|s2 )
(1
+
|h
c=1 j=1 k∈Z
06=µ∈D
ïÖsäÖ¥öçØã
X
F (t) =
(1 +
1
,
+ (ω/z)s2 )
z s1 )(1
Σ=
X
z = (h−1 |µj | cos φ) εt ,
F (k),
I=
Z
ω = h−2 c · 5−1/2 .
F (t) dt,
R
صñ%Û-éœÔ-ô δ = Σ − I ÷
È #$ /ɺ"#,™$]#HÎ$%$ÌÍs·ÂíÕ§ÞØã I õéºå2Õä2Ý
êè¥Ý
ãlÕ§Þ%ÛÒñÕ§×Õ§ãlÕ§ÞÖ-æ
k∈Z
1
I=
ln ε
Z
R+
1
dz
1
=
s
s
1
2
(1 + z )(1 + (ω/z) ) z
ln ε
Z
R+
t → z
z s2 −1
dz.
(1 + z s1 )(z s2 + ω s2 )
‚sƒ…„3†ˆ‡ w‰ „3ŠN‹ u Š„3‹ w~Œw1Ž ƒ Œ „ w1 „ ŒMu ‹ wG‘Hutj’wxŽ ƒ t<“ †”†
ÏléäØ
s2 > s 1 > 1
õ%Ô-Ö
1
I6
ln ε
ÏléäØÒö›Õ
s2 = s 1
õ%Ô-Ö
>1
I=
Z
z s2 −s1 −1
dz = C1 ω −s1 ,
z s2 + ω s2
C1 =
1
π
.
ln ε s2 sin (πs1 /s2 )
R+
1 −s1
1 ln ω
=
ω (ln ω + C2 ),
s
ln ε ω 1 − 1
ln ε
È #$ÑÐɺ"#,™$]HÎlÒ]2-”ÌÍ8·ÂíÕ§ÞØã
|δ| 6
Z
δ
0 < C2 < 1.
õØé§ñÖsäô
è§Û
ëÞÕ§×Ý
ê-Õ§ÞéœÔ-ê-Ö
þ µ
|F 0 (t)| dt.
R
¸ïãlÕ§Õ§ã
0
F (t) = − ln ε
Ö
Ô-Ù%Û°å+Ý
ïÖ-ùœÔ-Ö-ãqÛâñ×Ø
ÏléäØ
nµ
s2 > s 1
s2 = s 1 > 1
s2 (ω/z)s2
s1 z s1
−
1 + z s1
1 + (ω/z)s2
|F 0 (t)| 6 C3 F (t),
ñÖsäÛªìØã
>1
= s1
R+
F (t),
C3 = (s1 + s2 ) ln ε.
|δ| 6 C3 I 6 C1 C3 ω −s1 .
õ%Ô-Ö
Z
Z
|F 0 (t)| dt =
R
Z
|z − (ω/z) |
dz
(ω/z)s1
dz
6
2s
6
1
s
s
2
s
s
2
1
1
1
1
[(1 + z )(1 + (ω/z) )] z
[(1 + z )(1 + (ω/z) )] z
R+
Z
−s1
(ω/z)
dz
6 2s1
= 2ω −s1 .
s
2
(1 + z 1 ) z
s1
s1
R+
Ó äÕ¹å2Ö-êÝsÔ-Õäô-ÞÖõ
|δ| 6 2ω −s1 .
È #$ÔÑÉCÕ#$,*ÖÚ&”ÌÍT¸ïãlÕ§Õ§ã
ï×Ø
ω −s1 = c−s1 h2s1 · 5s1 /2 ,
s2 > s 1 > 1
ñÖsäÛªìÝ
Õ§ã
ln ω = 2 ln h−1 + ln c −
S(h, Mφ , s1 , s2 ) = 2
Q(c)
∞ X
X
ln 5
.
2
(I + δ) 6
c=1 j=1
6 2(C1 + C1 C3 )
Q(c)
∞ X
X
c=1 j=1
ω −s1 = C4 h2s1 ,
C4 = 2C1 (1 + C3 )5s1 /2 ζK (s1 ).
oqpGoqpsrt3uNvwx
n×
ï×Ø
õ2òºå2Õ Ö-ñ×Õ¹å2Õäë՜Ô-é£ë]Ø%è×Ý
ê-Õ§ÞéœÔ-êÝ
õé§Ö-Ö
Ô-ê-՜Ô-éœÔ-ê
Ûªý_ûÕ§Õgéä2Ý
òÝ
Õ§ãlÖ-Õ!éœÛªãlãlß
Ü Û°å2ÕœÔ k =ω[t]−s õñÖ-tùœÔ-Ö-ãqÛµñÖsäÛªìÕ§ÞÞÝsë¡Ö-íÕ§ÞÙªÝgëêsäzë=՜Ô-é£ωëÔ-Ö-ìÞÖ-æÎñÖçñÖ-×ë#å2Ù%Û-÷ÏléäØ s = s > 1Σõ
2
1
Ô-ÖgØãlÕ§Õ§ã
1
S(h, Mφ , s1 , s2 ) = 2
òºå2ÕaÙ#Ö-ÞéœÔsÝ
Þ%ÔsÝ
Ø
Q(c)
∞ X
X
(I + δ) = C ∗ h2s1 (ln h−1 + C5 ),
c=1 j=1
Ö
ñ
×
¹
Õ
2
å

Õ
ä
§
Õ
Þ
g
Ý
ê
©
l
¹õ
a
Ý
åäë C é§ñ×Ý
ê-Õ¹åäØê-Ögñ×Õ¹å2éœÔsÝ
êsäÕ§ÞØÕ
C∗
þ
5
−4 ln ε − ln 5
2 + 4 ln ε − ln 5
<∆<
.
4
4
$ï#$ͶzaäëâØ%è§ÛªìÕ§ÞØ%ëÝ
é§Øãlñ%Ô-Ö
Ô-ØÙØéœÛªãlãxÔ-ØñÝ S(h, M , s , s ) ãlÖ¥öçÞ֛ñ×ØêsäÕ§ÙªÝsÔ-ô
φ 1 2
C5 =
λK (s1 )
+ ∆,
2
؆å2×%Ûªò§ØÕ¡ãl՜Ô-Ö§å2ßïõÇÞÝ
ñ×ØãlÕ§× ãl՜Ô-Ö§å2ß Ô-Õ§Ö-×ØØYå2ØÖ-áaÝ
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è§Ûªý_ûØÕ
Ý
ññÝ
×ÝsÔgíÕ§ñÞßàÚå2×Ö-Ü-Õ§Ù
æ %Tõ Úh›ò‹ä*÷ ÷
ÛÆ Ñ!¿ Ì ÁÃÒÌÐÏ!ÄÒÑ Ë
¿ Ì Ï!Á¿
nÜ
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ï×Õ¹åäÖ¥ö›Õ§ÞÞÝsëKê-ßÇóÕüé‹àÕ§ãÝîÖ-íÕ§ÞÙØKéœÛªãlã ñÖîÝ|äò§Õ§Ü-×Ý
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Ù#ÖõqÙªÝ
نéœÔsÝ
Þ՜Ôüê-Ø#å2ÞÖúØ%èµñ×Øê-Ö§å2ØãlÖ-ò§ÖÎå+Ý|äÕ§ÕÒñ×ØãlÕ§×Ý%õlÕ§Õâ×Õ°Ý|äØ%è¥Ý|ð
íØ%ëÒÔ-Õà%ÞØìÕ§é§ÙØÎéäÖ¥öçÞÕ§Õ
÷
ÂÛ-éœÔ-ô s , . . . , s TñÖsäÖ¥öçØ%Ô-Õäô-ÞßÐÕaìØéä2ÝØ s TتàÒé§×Õ¹å2ÞÕ§Õ!òÝ
×ãlÖ-ÞØìÕ§é§Ù#Ö-Õ 
n
1
n
s=
ïÖsäÖ¥öçØã
S(h, M, s1 , . . . , sn ) =
òºå2Õ M ÞÕ§Ù#Ö
Ô-Ö-×ÝsëÒ×՜ó՜Ô-ÙªÝgê
ùœÔ-Ö
Ôç×ë#å¡é‹àÖ§å2Ø%Ô-é£ë™õ+ñ×ØìÕ§ã
s−1
1
X
n
.
+ . . . + s−1
n
1+
|h−1 x
1
|ns1
1
,
+ . . . + |h−1 xn |nsn
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ñ×Øãlէיõì%Ô-Ö!åäë
Rn
06=x∈M
M = Zn
Ø
þ×
s>1
S(h, Zn , s1 , . . . , sn ) hns∗
ñ×Ø
õ™òºå2Õ
÷ Ó äÕ¹åÛªý_û›Ýsë]Ô-Õ§Ö-×Õ§ãÝÒÖ-Ü-Ö-Ü
û›Ý
՜Ե×՜è§Û¥äô¥ÔsÝsÔ×Ý
Ü-Ö
Ô-ß × ÞÝ nð­hãl→
Õ§×Þ0ßÐæéäÛªsì∗ Ý
=æ™÷ min {s1, . . . , sn}
#Î#ï$TÍÝÞlß1àEá s > 1 â M ãºäå1æqå àEç~èé â…êHëeìjä èí´èHîLïEèeðñ Rn ç ëjò á~ó ë ó åCòôõ÷ö˜â ß åCò D
ñø ëjòùå ñ å ï å ß1àqñ å1ù-ùëú/ë ø ëjò û
ð è ò ú/å1ìjä è â-ö…å ß1ç â˜õüö˜â ß åCò
ø äâ
h → 0ý
qÝ
ÙØã -Ö Ü-×Ýsè°Ö-ãÐõñÖ-é§Ù#Ösäô-Ù%Û
K
ß1à å ø 1å ù-â
n ýþ-ëú3ÿ
è
S(h, M, s1 , . . . , sn ) hns
è¥ÝµØé§Ùªäý[ìÕ§ÞØÕ§ã~éäÛªìÝsë s = . . . = s õ2åäë Ý|äò§Õ§Üsð
þ
×Ý
ØìÕ§é§ÙتàÎ×՜ó՜Ô-Ö-Ù M éœÛªãlãÝ þ × ÛªÜ-ßÐêÝ
՜ÔâÙÞ%Û¥äýJÜ-ßÐéœÔ-×Õ§Õ
õ+ìÕ§ã†å1äëéœÔsÝ
Þ#å+Ý
×%nÔ-ÞÖ-æ¡×՜ó՜Ô-ÙØ
÷
M = Zn
ï×Õ¹å2êÝ
×Ø%Ô-Õäô-Þ֛å2Ö-ÙªÝsö›Õ§ãYÖ§å2ÞÖê-é§ñÖ-ãlÖ-òÝsÔ-Õäô-ÞÖ-ÕaÛ#Ô-ê-Õ§×ö›å2Õ§ÞØÕ
÷
#ïïT$ Í ù à åJú ä è ò
Z
Ru
+
s > s∗
1
w1 + . . . + w u +
ß õëeÿâ àqßjðüø äâ*ò î ìôõ ø ëjòë í â à åCò á ù-ôõ
Qu
−αl
l=1 wl
α1 , . . . , αu ý
dw1 . . . dwu
Qu
l=1 wl
‚sƒ…„3†ˆ‡ w‰ „3ŠN‹ u Š„3‹ w~Œw1Ž ƒ Œ „ w1 „ ŒMu ‹ wG‘Hutj’wxŽ ƒ t<“ †”†
n
],™…$ Õ#$2-'sÍ­zÖ-éœÔsÝsÔ-Ö-ìÞÖçå2Ö-ÙªÝsè¥ÝsÔ-ôõ+ì%Ô-Ögåäë¡ÞÕ§Ù#Ö
Ô-Ö-×ßà
Ø
êÚÙªÝsö›å2Ö-æ
Ø%èaÖ-Üsä2Ý
éœÔ-Õ§æê-Ø#å+Ý {(w , . . . , w ) ∈ Ru : w ≷ 1, . . . , w ≷ 1} ê-ßÐñÖsäÞÕ§ηÞ>ÖçÞ0Õ§×Ý
Cê-Õ§Þ>éœÔ-0ê-Ö
1
u
1
+
w1 + . . . + w u +
u
u
Y
wl−αl > Cw1±η . . . wu±η
l=1
ÞéœÔ-ê
Û w > 1 é§Ö-Ö
Ô-ê-՜Ô-éœÔ-ê
Û-՜ÔãlÞÖ¥öçØ%Ô-Õäô
þ ÞÕ§×Ý
ê-Õ§T
ãlÞÖ¥öçØ%l Ô-Õäô −η ÷ÂÛ-éœÔ-ôõÞÝ
ñ×Øãlէיõ
wl 6 1
wl
w1 > 1
Ö-òºå+Ý
w1 + . . . + w u +
õ÷§÷§÷§õ
u
Y
wl+η
êçñ×Ö-Ø%è°ê-Õ¹å2Õ§ÞØØ¡é§ñ×Ý
êÝ%õ2ÝçÞÕ§×Ý
ê-Õ§ÞéœÔ-ê
Û
õ
wl > 1 0 < wl+1 6 1
õ™÷§÷§÷§õ
0 < wu 6 1.
wl−αl > w1 + . . . + wl + (w1 . . . wl )−α (wl+1 . . . wu )−β >
l=1
òºå2Õ
> (w1 . . . wl )γ + (w1 . . . wl )−α (wl+1 . . . wu )−β = xγ + x−α y −β ,
õ
õ
α = max {α1 , . . . , αl } β = min {αl+1 , . . . , αu } γ = 1/l
·ÂéœÔsÝ|äÖ-é§ôÛªÜ-Õ¹å2Ø%Ô-ô-é£ë™õ2ì%Ô-Ö
êÖ-Üsä2Ý
éœÔ-Ø
¸ïãlÕ§Õ§ã
x = w1 . . . wl > 1,
õ
÷-ªÝ
é§é§ãlÖ
Ô-×Øã áïÛªÞÙíØý
f (x) = y η xγ−η + y −β+η x−α−η .
õòºå2Õ
x0 =
qÝ
ÙÒÙªÝ
Ù
δ
Ô-Ö
0 < y = wl+1 . . . wu 6 1.
xγ + x−α y −β > Cxη y −η
x>1 0<y61
f (x) > f (x0 )
õ
f (x0 ) y ,
α+η
γ−η
1/(α+γ)
y −β/(α+γ) .
βγ
β
δ=−
+η 1+
,
α+γ
α+γ
ñ ×Ø 0 < y 6 1 õÕ§éäØ η å2Ö-éœÔsÝsÔ-Ö-ìÞÖçãÝ|äÖ÷
],™$…Õ#$2-'ü#Î#ï!̀¸ïãlÕ§Õ§ã
f (x0 ) > C
S(h, M, s1 , . . . , sn ) =
å2Õ§é§ô
Q(c)
∞ X
X
X
1
=
=
2
F (k1 , . . . , ku ).
1 + |h−1 σ1 (µ)|ns1 + . . . + |h−1 σn (µ)|nsn
c=1 j=1 k∈Zu
06=µ∈D
X
F (y1 , . . . , yu ) =
1 + z1ns1
1
,
+ . . . + znnsn
zi = h−1 |σi (µj )|
u
Y
l=1
|σi (εl )|yl ,
1 6 i 6 n.
oqpGoqpsrt3uNvwx
nn
Óê-Õ¹å2Õ§ãYÖ-Ü-Ö
è°ÞÝ¥ìÕ§ÞØÕ
ω=
Ö-òºå+Ý%õêìÝ
éœÔ-ÞÖ-éœÔ-ؙõ
n
Y
zi = h−n c.
i=1
zn = ω
÷-ïÖsäÖ¥öçØã
u=n−1
þ ÞÝ
ñÖ-ãlÞØãÐõì%Ô-Ö
Σ=
X
u
Y
l=1
F (k1 , . . . , ku ),
I=
k∈Zu
∂zl
= ρml zl ,
∂ym
J
è¥Ý
ãlÕ§ÞßKñÕ§×Õ§ãlÕ§ÞÞßà
u
Y
zl ,
l=1
I=R
−1
Z
Ru
+
òºå2Õ
ρml = ln |σl (εm )|,
1+
Z
z1ns1
Ie =
Ru
+
ïÖ-éäÕïè¥Ý
ãlÕ§Þß ñÕ§×Õ§ãlÕ§ÞÞßà
+ ω
+...+
zunsu
1
dz1 . . . dzu
.
Qu −1 nsn Qu
+ ω l=1 zl
l=1 zl
(z1 , . . . , zu ) → (w1 , . . . , wu )
1
Qu
u
l=1
dz1 . . . dzu
e
Qu
6 R−1 I,
z
l
l=1
+...+
ñÖsäÛªìØã
n
−1 nsn
l=1 zl
Qu
zunsu
wl = ω −s zlnsl ,
Ie = ω −s
ØãlÕ§Õ§ã
R = | det (ρml )|.
1
z1ns1
F (y1 , . . . , yu ) dy1 . . . dyu ,
(y1 , . . . , yu ) → (z1 , . . . , zu )
J=R
Ó äÕ¹å2Ö-êÝsÔ-Õäô-ÞÖõ
Z
Ru
صñ%Û-éœÔ-ôõÙªÝ
ٵصê-ßÇóÕ
õ δ = Σ − I ÷
÷ ·ÂíÕ§ÞØãYØÞ%Ô-Õ§ò§×Ý|ä I ÷NqÝ
ÙÒÙªÝ
Ù
åäëÒëÙ#Ö-Ü-ØÝ
ÞÝ
zl−1
sl
Z
Ru
+
õòºå2Õ
1 6 l 6 u,
1
w1 + . . . + w u +
Qu
−sn /sl
l=1 wl
dw1 . . . dwu
Qu
.
l=1 wl
ïÖäÕ§ãlãlÕcñÖ-éäÕ¹å2ÞØæØÞ%Ô-Õ§ò§×Ý|äúé‹àÖ§å2Ø%Ô-é£ëÎؙõÔsÝ
ÙØãîÖ-Ü-×Ýsè°Ö-ãÐõé§ñ×Ý
ê-Õ¹åäØêÝÖ-íÕ§ÞÙªÝ
I 6 C1 ω −s .
y[÷N·ÂñØ×Ýsëé§ô!ÞÝ!ãlÞÖ-ò§Ö-ãlÕ§×ÞßÐæâÝ
ÞÝ|äÖ-ò[ÞÕ§×Ý
ê-Õ§ÞéœÔ-êÝ þ µ þ é§ãÐ÷Éõ#ÞÝ
ñ×ØãlէיõT õ#ãlÖ¥öçÞÖ!ñÖ-ÙªÝ|ð
è¥ÝsÔ-ôõì%Ô-Ö δ 6 C I ÷ Ó äÕ¹å2Ö-êÝsÔ-Õäô-ÞÖõ
2
Σ = I + δ 6 C3 ω −s .
‚sƒ…„3†ˆ‡ w‰ „3ŠN‹ u Š„3‹ w~Œw1Ž ƒ Œ „ w1 „ ŒMu ‹ wG‘Hutj’wxŽ ƒ t<“ †”†
Ó[÷-qÝ
ÙÒÙªÝ
Ù
ω −s = hns c−s
n
õÔ-Ö
S(h, M, s1 , . . . , sn ) 6 C3
∞
X
Q(c)c−s hns = C4 hns .
c=1
·ÂéœÔsÝ|äÖ-é§ôañÖ-ÙªÝsè¥ÝsÔ-ôõì%Ô-ÖañÖsäÛªìÕ§ÞÞÝsëçÖ-íÕ§ÞÙªÝïëêsäë՜Ô-é£ëÔ-Ö-ìÞÖ-ægñÖañÖ-×ë#å2Ù%Û-÷˜.Ô-ÖcÔsÝ
ٙõ#ñÖ-é§Ù#Ösäôsð
Ù%ÛâéœÛªãlãÝ Σ é§Ö§å2Õ§×öçØ%Ôçéä2Ý
òÝ
Õ§ãlÖ-Õ ω−s ÷˜z Õ§æéœÔ-ê-Ø%Ô-Õäô-ÞÖõ2ñ%Û-éœÔ-ô y õ÷§÷§÷§õ y ÔsÝ
Ù#Ö-ê-ßïõì%Ô-Ö
1
Ö-òºå+Ý
zlnsl = ω s ,
u
1 6 l 6 u.
Ø ãlÖ¥öçÞÖê
è§ë%Ô-ô k = [y ] õ÷§÷§÷§õ k = [y ] ÷
µ
1
1
u
u
Ó~è¥Ý
Ùªäý[ìÕ§ÞØÕçñ×Øê-Õ¹å2Õ§ã~՜ûÕgÖ§å2ØÞüñ×ØãlÕ§×]ÔsÝ
Ù#Ö-ò§Ö×Ö§å+ݵÖ-íÕ§ÞÙØ þ ×Ý
é§ñ×Ö-éœÔ-×Ý
ÞÕ§ÞØÕÚñ×تð
ãlÕ§×ÝØ%èa×Ýsèå÷
ÞÝ nð­ãlÕ§×ÞßÐæµéäÛªìÝ
æ 
F (y1 , . . . , yu ) ω −s
X
06=x∈M
1
(1 +
|h−1 x
1
|s1 ) . . . (1
+
|h−1 x
n
|s n )
hns∗ (ln h−1 )v−1 ,
h → 0,
ºò å2Õ s = min {s , . . . , s } > 1 õ2Ý v ìØéäÖçÞÖ-ãlÕ§×Ö-ê i õåäëÙ#Ö
Ô-Ö-×ßà s = s ÷ï×Ø v = n ÛªÙªÝ|ð
è¥Ý
ÞÞÝs∗ëâÖ-íÕ§ÞÙªÝcå21 Ö-ñ%Û-é§ÙªÝ
n՜ÔÛ#Ô-Ö-ìÞÕ§ÞØÕ þ é§ãÐ÷Ý
ÞÝ|äÖ-ò§ØìÞßÐæÒ×՜è§Û¥äô¥ÔsÝsÔçê* i õ%é§ác∗Ö-×ãqÛ¥äØ×Ö-êÝ
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åäëµéœÛªãlãlß (∗) ÷ ÝÞÕ¹å2Ö-éœÔsÝsÔ-Ù#Ö-ã ãlÕ§éœÔsÝå2Ö-ÙªÝsè¥ÝsÔ-Õäô-éœÔ-ê-ÖçùœÔ-تàÒÛ#Ô-ê-Õ§×ö›å2Õ§ÞØæÎãlß Ö-ñ%Û-é§ÙªÝ
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