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PureMathematics
n
Ø
ê
Æ
,2021,11(4),442-453
PublishedOnlineApril2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.114057
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Ramanujan
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Á
‡
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z
õ
c
c
,Ramanujan
Ä
g
½
Â
²
;
Ramanujan
Ú
:
c
q
(
n
) =
X
k
mod
q
(
k,q
)=1
e
2
πikn
q
(
q,n
∈
N
)
,
Ù
¥
N
´
ê
8
.
(
k,q
)
´
k
Ú
q
•
Œ
ú
Ï
f
.1976
c
,
3
Wintner
(
J
Ä
:
þ
§
Delange
y
²
½
Â
3
ê
‚
Z
þ
ü
C
þ
Ž
â
¼
ê
Œ
±
Ï
L
Ramanujan
Ú
\
±
Ð
m
. 2018
c
, T´oth
y
²
½
Â
3
Z
þ
õ
Ž
â
¼
ê
Œ
±
Ï
L
Ramanujan
Ú
†j
Ramanujan
Ú
\
±
Ð
m
.
3
d
Ä
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þ
,
©
Á
ã
ò
½
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3
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ê
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õ
n
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¼
ê
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Ramanujan
Ú
\
±
Ð
m
,
Ó
ž
•
ò
?
˜
Ú
ï
Ä
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ê
ê
‚
þ
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Ú
¦
5
†
'
X
5
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RamanujanExpansionoverAlgebraic
IntegerRings
XuruiLiu
SouthChinaUniversityofTechnology,GuangzhouGuangdong
Email:lxr19927525199@163.com
©
Ù
Ú
^
:
4
R
a
.
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Ramanujan
Ð
m
[J].
n
Ø
ê
Æ
,2021,11(4):442-453.
DOI:10.12677/pm.2021.114057
4
R
a
Received:Mar.4
th
,2021;accepted:Apr.6
th
,2021;published:Apr.13
th
,2021
Abstract
One hundred yearsago, Ramanujan firstdefined thefollowing classicRamanujan sum:
c
q
(
n
) =
X
k
mod
q
(
k,q
)=1
e
2
πikn
q
(
q,n
∈
N
)
,
where
N
isthesetofpositiveintegers,and
(
k,q
)
isthegreatestcommonfactorof
k
and
q
.In1976,onthebasisofWintner’sresults,Delangeprovedthatallunivariate
arithmeticfunctionsdefinedontheintegerring
Z
canbeexpandedbyRamanujan
sum.In2018,T´othprovedthatthemultivariatearithmeticfunctiondefinedon
Z
canbeexpandedbyRamanujansumandunitaryRamanujansum.Onthisbasis,
thispaperattemptstoexpandthemultivariateidealfunctiondefinedon
D
through
theRamanujansum.Atthesametime,itwillfurtherstudythemultiplicativeand
orthogonalrelationsoftheRamanujansumon
D
.
Keywords
RamanujanSum,
D
,MultivariateIdealFunction
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.
Ú
ó
˜
z
õ
c
c
,Ramanujan
Ä
g
½
Â
X
e
²
;
Ramanujan
Ú
:
c
q
(
n
) =
X
k
mod
q
(
k,q
)=1
e
2
πikn
q
(
q,n
∈
N
)
,
(1.1)
Ù
¥
N
´
ê
8
,(
k,q
)
´
k
Ú
q
•
Œ
ú
Ï
f
.
1976
c
, Delange [1]
3
Wintner [2]
(
J
Ä
:
þ
y
²
½
Â
3
ê
‚
Z
þ
ü
C
þ
Ž
â
¼
ê
DOI:10.12677/pm.2021.114057443
n
Ø
ê
Æ
4
R
a
Ñ
Œ
±
Ï
L
Ramanujan
Ú
\
±
Ð
m
.
ù
a
q
u
²
;
ê
Æ
©
Û
¥±
Ï
¼
ê
Fourier
Ð
m
ª
.
¦
(
J
X
e
:
½
n
1
(Delange[1])
.
f
:
N
→
C
´
?
¿
Ž
â
¼
ê
.
X
J
∞
X
n
=1
2
ω
(
n
)
|
(
µ
∗
f
)(
n
)
|
n
<
∞
.
@
o
é
?
¿
n
∈
N
§
·
‚
k
e
¡
ý
é
Â
ñ
Ramanujan
Ð
m
ª
f
(
n
) =
∞
X
q
=1
a
q
c
q
(
n
)
,
Ù
¥
X
ê
a
q
d
e
ª
‰
Ñ
a
q
=
∞
X
m
=1
(
µ
∗
f
)(
mq
)
mq
(
q
∈
N
)
.
,
, Delange
„
±
þ
(
J
é
u
¦
5
¼
ê
A^
.
I
‡
•
Ñ
´
,
3
Delange
ƒ
c
, Cohen
[3]
•
Q
é
,
A
Ï
ü
C
þ
¦
5
¼
ê
a
í
Ñ
ý
é
Â
ñ
Ramanujan
Ð
m
ª
•{
.
2016
c
, Ushiroya [4]
ò
þ
ã
(
J
í
2
ü
‡
C
þ
œ
/
,
Ó
ž
½
Â
3
Z
þ
,
A
Ï
¼
ê
²
;
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Ð
m
ª
ä
N
L
ˆ
.
2018
c
,T´oth[5]
q
ò
Ù
í
2
õ
œ
/
,
y
²
½
Â
3
Z
þ
õ
Ž
â
¼
ê
Ñ
Œ
±
Ï
L
Ramanujan
Ú
\
±
Ð
m
.T´oth
‰
Ñ
±
e
½
Â
:
é
?
¿
½
k
∈
N
,
-
f,g
:
N
k
→
C
´
ü
‡
Ž
â
¼
ê
,
@
o
§
‚
Dirichlet
ò
È
½
Â
•
(
f
∗
g
)(
n
1
,
···
,n
k
) =
X
d
1
|
n
1
,
···
,d
k
|
n
k
f
(
d
1
,
···
,d
k
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g
n
1
d
1
,
···
,
n
k
d
k
.
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¿
±
e
½
n
.
½
n
2
(T´oth[5]
½
n
2)
.
b
f
:
N
k
→
C
´
?
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Ž
â
¼
ê
(
k
∈
N
).
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J
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X
n
1
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···
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k
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2
ω
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n
1
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···
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n
k
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|
(
µ
k
∗
f
)(
n
1
,
···
,n
k
)
|
n
1
···
n
k
<
∞
.
(1.3)
@
o
,
é
?
¿
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1
,
···
,n
k
∈
N
,
·
‚
k
ý
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ñ
Ramanujan
Ð
m
ª
f
(
n
1
,
···
,n
k
) =
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q
1
,
···
,q
k
=1
a
q
1
,
···
,q
k
c
q
1
(
n
1
)
···
c
q
k
(
n
k
)
,
(1.4)
Ù
¥
a
q
1
,
···
,q
k
=
∞
X
m
1
,
···
,m
k
=1
(
µ
k
∗
f
)(
m
1
q
1
,
···
,m
k
q
k
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m
1
q
1
···
m
k
q
k
.
3ù
‡
(
J
¥
-
f
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n
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k
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g
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n
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k
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,
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k
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k
∈
N
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f
,
g
•
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→
C
?
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â
¼
ê
, T´oth
?
˜
Ú
±
e
(
Ø
.
DOI:10.12677/pm.2021.114057444
n
Ø
ê
Æ
4
R
a
½
n
3
(T´oth[5]
½
n
3)
.
b
f
:
N
→
C
?
˜
Ž
â
¼
ê
(
k
∈
N
).
X
J
∞
X
n
=1
2
kω
(
n
)
|
(
µ
∗
f
)(
n
)
|
n
k
<
∞
.
(1.5)
@
o
,
é
?
¿
n
1
,
···
,n
k
∈
N
,
·
‚
k
e
¡
ý
é
Â
ñ
?
ê
g
((
n
1
,
···
,n
k
)) =
∞
X
q
1
,
···
,q
k
=1
a
q
1
···
,q
k
c
q
1
(
n
1
)
···
c
q
k
(
n
k
)
,
(1.6)
g
((
n
1
,
···
,n
k
)) =
∞
X
q
1
···
,q
k
=1
a
∗
q
1
···
,q
k
c
∗
q
1
(
n
1
)
···
c
∗
q
k
(
n
k
)
,
(1.7)
Ù
¥
a
q
1
,
···
,q
k
=
1
Q
k
∞
X
m
=1
(
µ
∗
f
)(
mQ
)
m
k
,
(1.8)
a
∗
q
1
,
···
,q
k
=
1
Q
k
∞
X
m
=1
(
m,Q
)=1
(
µ
∗
f
)(
mQ
)
m
k
.
…
Q
= [
q
1
,
···
,q
k
].
5
¿
¦
5
¼
ê
Œ
±
d
Ù
3
ƒ
˜
?
Š
(
½
§
d
d
(
J
„
Œ
?
˜
Ú
Z
þ
,
A
Ï
õ
¦
5
¼
ê
'
u
²
;
Ramanujan
Ú
±
9
j
Ramanujan
Ú
Ð
m
ª
ä
N
L
ˆ
§
•
?
˜
Ú
/
§
§
‚
†
²
;
Riemannzeta
¼
ê
ζ
(
z
)
k
'
.
A
+
´
k
•
•
þ
˜
õ
‘
ª
‚
A
=
F
q
[
T
]
¥
Ä
˜
õ
‘
ª
N
.
a
'
²
;
Ramanujan
Ú
,
{
I
ê
Ø
Æ
[
L.Carlitz[6]
Ä
g
Ú
\
A
þ
H
õ
‘
ª
Ramanujan
Ú
η
(
G,H
)
½
Â
µ
η
(
G,H
) =
X
D
mod
H
(
D,H
)=1
E
(
G,H
)(
D
)
,
Ù
¥
G,H
∈
A
,(
G,H
)
´
G,H
Ä
˜
•
Œ
ú
Ï
f
.
·
I
ê
Ø
Æ
[
x“
]
[7]
Ç
3
•
C
©
z
¥
X
Ú
/
ï
Ä
õ
‘
ª
Ramanujan
Ú
5
Ÿ
.
a
'
j
Ramanujan
Ú
½
Â
,
à
X
•
3
¨
.
’
Ø
©
[8]
¥
½
Â
A
þ
j
õ
‘
ª
Ramanujan
Ú
η
∗
(
G,H
)
µ
η
∗
(
G,H
) =
X
D
mod
H
(
D,H
)
∗
=1
E
(
G,H
)(
D
)
.
(1.9)
Ù
¥
G,H
∈
A
,(
G,H
)
∗
= max
deg
{
D
:
D
|
H,D
||
G
}
,
=
(
G,H
)
∗
´
õ
‘
ª
8
Ü
{
D
:
D
|
H,D
||
G
}
¥
g
ê
•
p
.
DOI:10.12677/pm.2021.114057445
n
Ø
ê
Æ
4
R
a
¨
•
ò
þ
ã
k
'
Ramanujan
Ð
m
(
J
?
˜
Ú
í
2
k
•
•
˜
õ
‘
ª
‚
A
¥
§
½
Â
3
A
þ
õ
Ž
â
¼
ê
Ñ
Œ
±
Ï
L
õ
‘
ª
Ramanujan
Ú
±
9
j
õ
‘
ª
Ramanujan
Ú
\
±
Ð
m
.
½
n
4
(
à
X
•
[8]
½
n
7)
.
-
f
: (
A
+
)
k
→
C
?
˜
Ž
â
¼
ê
§
Ù
¥
k
∈
N
.
X
J
X
G
1
,
···
,G
k
∈
A
+
2
ω
(
G
1
)+
···
+
ω
(
G
k
)
|
(
µ
k
∗
f
)(
G
1
,
···
,G
k
)
|
|
G
1
|···|
G
k
|
<
∞
.
(1.10)
@
o
,
é
?
¿
G
1
,
···
,G
k
∈
A
+
,
·
‚
k
ý
é
Â
ñ
Ramanujan
Ð
m
ª
f
(
G
1
,
···
,G
k
) =
X
H
1
,
···
,H
k
∈
A
+
C
H
1
,
···
,H
k
η
(
G
1
,H
1
)
···
η
(
G
k
,H
k
)
,
(1.11)
Ú
f
(
G
1
,
···
,G
k
) =
X
H
1
,
···
,H
k
∈
A
+
C
∗
H
1
,
···
,H
k
η
∗
(
G
1
,H
1
)
···
η
∗
(
G
k
,H
k
)
,
(1.12)
Ù
¥
C
H
1
,
···
,H
k
=
X
M
1
,
···
,M
k
∈
A
+
(
µ
k
∗
f
)(
M
1
H
1
,
···
,M
k
H
k
)
|
M
1
H
1
|···|
M
k
H
k
|
,
C
∗
H
1
,
···
,H
k
=
X
M
1
,
···
,M
k
∈
A
+
(
M
1
,H
1
)=1
,
···
,
(
M
k
,H
k
)=1
(
µ
k
∗
f
)(
M
1
H
1
,
···
,M
k
H
k
)
|
M
1
H
1
|···|
M
k
H
k
|
.
(1.13)
-
(
G
1
,
···
,G
k
)
†
[
G
1
,
···
,G
k
]
©
O
•
G
1
,
···
,G
k
∈
A
•
Œ
ú
Ï
f
9
•
ú
ê
.
g
´
l
A
+
C
Ž
â
¼
ê
.
-
½
n
7
¥
f
(
G
1
,
···
,G
k
)=
g
((
G
1
,
···
,G
k
)),
¨
q
?
˜
Ú
e
¡
(
J
.
½
n
5
(
à
X
•
[8]
½
n
8)
.
]
-
g
:
A
+
→
C
?
˜
Ž
ê
¼
ê
,
k
∈
N
.
X
J
X
G
∈
A
+
2
kω
(
G
)
|
(
µ
∗
g
)(
G
)
|
|
G
|
k
<
∞
.
(1.14)
@
o
é
?
¿
G
1
,
···
,G
k
∈
A
+
,
k
ý
é
Â
ñ
?
ê
g
((
G
1
,
···
,G
k
)) =
X
H
1
,
···
,H
k
∈
A
+
C
H
1
,
···
,H
k
η
(
G
1
,H
1
)
···
η
(
G
k
,H
k
)
,
(1.15)
Ú
g
((
G
1
,
···
,G
k
)) =
X
H
1
,
···
,H
k
∈
A
+
C
∗
H
1
,
···
,H
k
η
∗
(
G
1
,H
1
)
···
η
∗
(
G
k
,H
k
)
,
(1.16)
Ù
¥
C
H
1
,
···
,H
k
=
1
|
Q
|
k
X
M
∈
A
+
(
µ
∗
g
)(
MQ
)
|
M
|
k
,
C
∗
H
1
,
···
,H
k
=
1
|
Q
|
k
X
M
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+
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M,Q
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g
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DOI:10.12677/pm.2021.114057446
n
Ø
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4
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a
…
Q
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1
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···
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k
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DOI:10.12677/pm.2021.114057447
n
Ø
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4
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DOI:10.12677/pm.2021.114057448
n
Ø
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Æ
4
R
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|
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|
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(
p
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,
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p
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|
m
0
,
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§
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,
X
J
p
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|
m
,
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d
|
p
e
C
(
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,
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=
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m
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C
(
m
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p
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p
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p
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(
p
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−
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=
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(
p
)
e
.
DOI:10.12677/pm.2021.114057449
n
Ø
ê
Æ
4
R
a
X
J
p
i
-
m
,
p
i
−
1
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|
p
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C
(
m
,
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=
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m
,O
k
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(
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,
p
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p
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−
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p
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−
1
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−
N
(
p
i
−
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N
(
p
i
−
1
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= 0
.
·
‚
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|
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(
m
,
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=
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(
m
,
d
s
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=
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(
p
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1
|
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(
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−
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,
DOI:10.12677/pm.2021.114057450
n
Ø
ê
Æ
4
R
a
ª
(2.3)
¤
á
,
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ž
·
‚
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±
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|
p
e
|
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(
m
,
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(
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DOI:10.12677/pm.2021.114057451
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