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PureMathematicsnØêÆ,2021,11(4),442-453
PublishedOnlineApril2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.114057
“êê‚þRamanujanÐm
444RRRaaa
uHnóŒÆ§2À2²
Email:lxr19927525199@163.com
ÂvFϵ2021c34F¶¹^Fϵ2021c46F¶uÙFϵ2021c413F
Á‡
Œ˜zõcc,RamanujanÄg½Â²;Ramanujan Ú:
c
q
(n) =
X
kmodq
(k,q)=1
e
2πikn
q
(q,n∈N),
Ù¥N´ê8.(k,q)´kÚq•ŒúÏf.1976c,3Wintner(JÄ:þ§
Delange y²½Â3ê‚ZþüCþŽâ¼êŒ±ÏLRamanujan Ú\±Ðm. 2018
c, T´oth y²½Â3ZþõŽâ¼êŒ±ÏLRamanujan Ú†jRamanujanÚ\±Ð
m. 3dÄ:þ, ©Áãò½Â3“êê‚þõnŽ¼êÏLRamanujan Ú\±Ðm,
Óž•ò?˜ÚïÄ“êê‚þRamanujanÚ¦5†'X5Ÿ.
'…c
RamanujanÚ,“êê‚,õnŽ¼ê
RamanujanExpansionoverAlgebraic
IntegerRings
XuruiLiu
SouthChinaUniversityofTechnology,GuangzhouGuangdong
Email:lxr19927525199@163.com
©ÙÚ^:4Ra.“êê‚þRamanujanÐm[J].nØêÆ,2021,11(4):442-453.
DOI:10.12677/pm.2021.114057
4Ra
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
kmodq
(k,q)=1
e
2πikn
q
(q,n∈N),
whereNisthesetofpositiveintegers,and(k,q)isthegreatestcommonfactorofk
andq.In1976,onthebasisofWintner’sresults,Delangeprovedthatallunivariate
arithmeticfunctionsdefinedontheintegerringZcanbeexpandedbyRamanujan
sum.In2018,T´othprovedthatthemultivariatearithmeticfunctiondefinedonZ
canbeexpandedbyRamanujansumandunitaryRamanujansum.Onthisbasis,
thispaperattemptstoexpandthemultivariateidealfunctiondefinedonDthrough
theRamanujansum.Atthesametime,itwillfurtherstudythemultiplicativeand
orthogonalrelationsoftheRamanujansumonD.
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.Úó
˜zõcc,RamanujanÄg½ÂXe²;RamanujanÚ:
c
q
(n) =
X
kmodq
(k,q)=1
e
2πikn
q
(q,n∈N),(1.1)
Ù¥N´ê8,(k,q) ´kÚq•ŒúÏf.
1976c, Delange [1]3Wintner [2](JÄ:þy²½Â3ê‚ZþüCþŽâ¼ê
DOI:10.12677/pm.2021.114057443nØêÆ
4Ra
ÑŒ±ÏLRamanujanÚ\±Ðm.ùa qu²;êÆ©Û¥±Ï¼êFourierÐmª.¦(
JXe:
½n1(Delange[1]).f: N→C´?¿Žâ¼ê.XJ
∞
X
n=1
2
ω(n)
|(µ∗f)(n)|
n
<∞.
@oé?¿n∈N§·‚ke¡ýéÂñRamanujanÐmª
f(n) =
∞
X
q=1
a
q
c
q
(n),
Ù¥Xêa
q
deª‰Ñ
a
q
=
∞
X
m=1
(µ∗f)(mq)
mq
(q∈N).
,, Delange „±þ(Jéu¦5¼êA^.I‡•Ñ´,3Delangeƒc, Cohen
[3]•Qé,AÏüCþ¦5¼êaíÑýéÂñRamanujanÐmª•{.
2016c, Ushiroya [4]òþã(Jí2ü‡ Cþœ/, Óž½Â3Zþ,Aϼê
²;RamanujanÐmªäNLˆ.
2018c,T´oth[5]qòÙí2õœ/,y²½Â3ZþõŽâ¼êÑŒ±ÏL
RamanujanÚ\±Ðm.T´oth‰Ñ±e½Â:
é?¿½k∈N,-f,g: N
k
→C´ü‡Žâ¼ê,@o§‚DirichletòȽ•
(f∗g)(n
1
,···,n
k
) =
X
d
1
|n
1
,···,d
k
|n
k
f(d
1
,···,d
k
)g

n
1
d
1
,···,
n
k
d
k

.(1.2)
¿±e½n.
½n2(T´oth[5]½n2).bf: N
k
→C´?˜Žâ¼ê(k∈N).XJ
∞
X
n
1
,···,n
k
=1
2
ω(n
1
)+···+ω(n
k
)
|(µ
k
∗f)(n
1
,···,n
k
)|
n
1
···n
k
<∞.(1.3)
@o,é?¿n
1
,···,n
k
∈N,·‚kýéÂñRamanujanÐmª
f(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
),(1.4)
Ù¥
a
q
1
,···,q
k
=
∞
X
m
1
,···,m
k
=1
(µ
k
∗f)(m
1
q
1
,···,m
k
q
k
)
m
1
q
1
···m
k
q
k
.
3ù‡(J¥-f(n
1
,···,n
k
)=g((n
1
,···,n
k
)),Ù¥(n
1
,···,n
k
)´n
1
,···,n
k
∈N•Œ
úÏf,g•N→C?˜Žâ¼ê, T´oth?˜Ú±e(Ø.
DOI:10.12677/pm.2021.114057444nØêÆ
4Ra
½n3(T´oth[5]½n3).bf: N→C?˜Žâ¼ê(k∈N).XJ
∞
X
n=1
2
kω(n)
|(µ∗f)(n)|
n
k
<∞.(1.5)
@o,é?¿n
1
,···,n
k
∈N,·‚ke¡ýéÂñ?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¿¦5¼êŒ± dÙ3ƒ˜?Š(½§dd(J„Œ?˜ÚZþ,AÏ
õ¦5¼ê'u²;RamanujanÚ±9jRamanujanÚÐmªäNLˆ§•?˜Ú/§§‚
†²;Riemannzeta¼êζ(z)k'.
A
+
´k••þ˜õ‘ª‚A= F
q
[T]¥Ä˜õ‘ªN.a'²;RamanujanÚ,{
IêØÆ[L.Carlitz[6]ÄgÚ\AþHõ‘ªRamanujanÚη(G,H)½Âµ
η(G,H) =
X
DmodH
(D,H)=1
E(G,H)(D),
Ù¥G,H∈A,(G,H)´G,HĘ•ŒúÏf.
·IêØÆ[x“][7]Ç3•C©z¥XÚ/ïÄõ‘ªRamanujanÚ5Ÿ.
a'jRamanujan Ú½Â,àX•3¨.’Ø©[8]¥½ÂAþjõ‘ªRamanujan
Úη
∗
(G,H)µ
η
∗
(G,H) =
X
DmodH
(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‘ª8Ü{D: D|H,D||G}¥gê
•p.
DOI:10.12677/pm.2021.114057445nØêÆ
4Ra
¨•òþãk'RamanujanÐm(J?˜Úí2k••˜õ‘ª‚A¥§½
Â3AþõŽâ¼êÑŒ±ÏLõ‘ªRamanujanÚ±9jõ‘ªRamanujanÚ\±Ðm.
½n4(àX•[8]½n7).-f: (A
+
)
k
→C?˜Žâ¼ê§Ù¥k∈N.XJ
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
+
,·‚kýéÂñRamanujanÐmª
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•ŒúÏf9•úê.g´l
A
+
CŽ â¼ê.-½n7 ¥f(G
1
,···,G
k
)=g((G
1
,···,G
k
)),¨q ?˜Úe¡
(J.
½n5(àX•[8]½n8).]-g: A
+
→C?˜Žê¼ê,k∈N.XJ
X
G∈A
+
2
kω(G)
|(µ∗g)(G)|
|G|
k
<∞.(1.14)
@oé?¿G
1
,···,G
k
∈A
+
,kýéÂñ?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∈A
+
(M,Q)=1
(µ∗g)(MQ)
|M|
k
,
(1.17)
DOI:10.12677/pm.2021.114057446nØêÆ
4Ra
…Q:= [H
1
,···,H
k
].
2020c§Œ'[9]3•C©z¥½Â“êê‚RamanujanÚµ
C(m,n) =
X
d|(m,n)
N(d)µ(n/d) =
X
x(modn)
(x,n)=1
e
2πiTr(xy)
,(1.18)
Ù¥m,n•“êê‚š"nŽ,y•ÌnŽhyi=mn
−1
D
−1
0
RÌÏf,…D
−1
0
={x∈
K|Tr(xD) ⊆Z},Tr(xy)•ƒxy,.
½n6(Œ'[9]½n1.2).]D•ê•K“êê‚, -n•“êê‚D¥?˜š"nŽ,
m•D¥?˜nŽ.@o•3˜‡nŽR÷v(R,mnD
0
) = 1…mn
−1
D
−1
0
R´˜‡©ªÌnŽ.
-y∈K•mn
−1
D
−1
0
R)¤,=hyi= mn
−1
D
−1
0
R.·‚k
C(m,n) =
X
x(modn)
(x,n)=1
e
2πiTr(xy)
,(1.19)
Ù¥
D
−1
0
= {x∈K|Tr(xD) ⊆Z},(1.20)
©ªnŽD
−1
0
´D/Z©D
0
Ö8,w,D
0
´“êê‚D¥nŽ.
©•Äò®½Â3ZþõŽâ¼êÏLRamanujan Ú\±Ðm, ?˜Úí2
½Â3“êê‚Dþõn Ž¼êŒ±ÏLRamanujanÚ\±Ðm,¿±e(J.
½n7.bf: D
k
→C•?˜õnŽ¼ê,(k∈N).XJ
X
m
1
,···,m
k
/D
2
ω(m
1
)+···+ω(m
k
)
|(µ
k
∗f)(m
1
,···,m
k
)|
N(m
1
)...N(m
k
)
<∞(1.21)
@o,é?¿m
1
,···,m
k
D,·‚ke¡ýéÂñ?ê
f(m
1
,···,m
k
) =
X
n
1
,···,n
k
/D
η
n
1
,···,n
k
C(m
1
,n
1
)···C(m
k
,n
k
),(1.22)
Ù¥
η
n
1
,···,n
k
=
X
l
1
,···,l
k
/D
(µ
k
∗f)(n
1
l
1
,···,n
k
l
k
)
N(n
1
l
1
)...N(n
k
l
k
)
.(1.23)
2.½n7y²9ÙíØ
•y²Ù(Ø,I‡e¡·K.é“êê‚D¥?˜š"nŽn,y∈K •
mn
−1
D
−1
0
R)¤Ïf.
·K1.e¡ù‡ª¤á
X
xmodn
e
2πiTr(xy)
=
(
N(n),en|m
0,Ù§
DOI:10.12677/pm.2021.114057447nØêÆ
4Ra
y²XJn|m,@omn
−1
∈D,Ï•hyi= mn
−1
D
−1
0
R⊆D
−1
0
,·‚Œ±y∈D
−1
0
,•
Ò´`Tr(xy)∈Z,e
2πiTr(xy)
= 1.Ïd,
X
xmodn
e
2πiTr(xy)
=N(n).
XJn-m,@o-
A=
X
xmodn
e
2πiTr(xy)
.
5¿,XJα´˜‡ê,Kx²L˜‡•{Xmodnž,x+α½,, •Ò´`
A=A·e
2πiTr(αy)
.(2.1)
z‡êα•êÏfØUu1,Ï•Tr(αy)o´˜‡knê, Ïd, ŠâD
−1
0
½Â, ·‚Œ±
n|m†bƒ‡. Ïdl(2.1)Œ±íÑA=0.
·K2.C(m,n)´'un∈D¦5¼ê.
y²-n
1
,n
2
•“êê‚D¥nŽ¿÷v(n
1
,n
2
)=1.,·‚Œ±
C(m,n
1
n
2
) =
X
x(modn
1
n
2
)
(x,n
1
n
2
)=1
e
2πiTr(xy)
=
X
x(modn
1
)
(x,n
1
)=1
e
2πiTr(xy
1
)
X
x(modn
2
)
(x,n
2
)=1
e
2πiTr(xy
2
)
= C(m,n
1
)C(m,n
2
),
Ù¥hy
1
i= mn
−1
1
D
−1
0
R, hy
2
i= mn
−1
2
D
−1
0
R,Úhy
1
y
2
i=hyi.
Ï•C(m,n) ûuÙƒnŽ˜.duC(m,n) ´¦5¼ê,·‚Œ±±e·K.
·K3.éuD¥?¿š"nŽq,m,k±eª¤á
X
d|n
C(m,d) =
(
N(n),en|m
0,Ù§
(2.2)
y²Äk,·‚5y²±eª¤á
C(m,p
e
) =







N(p)
e
−N(p)
e−1
,ep
e
|m
−N(p)
e−1
,ep
e
-m,p
e−1
|m
0,Ù§
DOI:10.12677/pm.2021.114057448nØêÆ
4Ra
XJp
e
|m,K
C(m,p
e
) =
X
x(modp
e
)
(x,p
e
)=1
e
2πiTr(xy)
.
dup
e
|mž,e
2πiTr(xy)
=1. ,
C(m,p
e
) =
X
x(modp
e
)
(x,p
e
)=1
1 = ϕ(p
e
) = N(p)
e
−N(p)
e−1
.
p
e
-mž, ·‚k±eœ/,
(m,p
e
) = p
e−1
, d= p
e−1
,···,p,O
k
(m,p
e
) = p
e−2
, d= p
e−2
,···,p,O
k
······
(m,p
e
) = 1, d= O
k
ùL«p
e
-m,p
e−1
|m,
C(m,p
e
) = −N(p)
e−1
,
3Ù¦œ¹e,C(m,p
e
) = 0.
e5,énŽn‰ƒ©),kn= p
e
1
1
...p
e
s
s
.·‚Œ±†
X
d|n
C(m,d)
=
X
d
1
...d
s
|p
e
1
1
...p
e
s
s
C(m,d
1
...d
s
)
=
X
d
1
|p
e
1
1
···
X
d
s
|p
e
s
s
C(m,d
1
...d
s
)
=
X
d
1
|p
e
1
1
C(m,d
1
)···
X
d
s
|p
e
s
s
C(m,d
s
).
·‚äó,
X
d|p
e
C(m,d) =
(
N(p)
e
,ep
e
|m
0,Ù§
¯¢þ,XJp
e
|m,@o
X
d|p
e
C(m,d)
= C(m,O
k
)+C(m,p)+···+C(m,p
e
)
= 1+(N(p)−1)+···+(N(p)
e
−N(p)
e−1
)
= N(p)
e
.
DOI:10.12677/pm.2021.114057449nØêÆ
4Ra
XJp
i
-m,p
i−1
|m,i= 1,···,e,@o
X
d|p
e
C(m,d)
= C(m,O
k
)+···+C(m,p
i−1
)+C(m,p
i
)+···+C(m,p
e
)
= 1+(N(p)−1)+···+(N(p
i−1
)−N(p
i−2
))+N(p
i−1
)+0
= 0.
·‚íä
X
d|n
C(m,d)
=
X
d
1
|p
e
1
1
C(m,d
1
)···
X
d
s
|p
e
s
s
C(m,d
s
)
=
(
N(p
1
)
e
1
,p
e
1
1
|m
0,Ù§
...
(
N(p
s
)
e
s
,p
e
s
s
|m
0,Ù§
=
(
N(n),en|m
0,Ù§.
½n7 y²·‚I‡ky²
X
d|p
e
|C(m,d)|=
(
N(p)
e
,ep
e
|m
2N(p)
i−1
,ep
i
-m,p
i−1
|m,i= 1,···,e
(2.3)
¯¢þ,XJp
e
|m,@oŒ±
X
d|p
e
|C(m,d)|
= |C(m,1)|+|C(m,p)|+···+|C(m,p
e
)|
= 1+|N(p)−1|+···+


N(p)
e
−N(p)
e−1


= 1+(N(p)−1)+···+(N(p)
e
−N(p)
e−1
)
= N(p)
e
,
XJp
i
-m,p
i−1
|m,i= 1,···,e,@o
X
d|p
e
|C(m,d)|= 2N(p)
i−1
,
DOI:10.12677/pm.2021.114057450nØêÆ
4Ra
ª(2.3) ¤á,Óž·‚Œ±
X
d|p
e
|C(m,d)|≤2N(p)
v
p
(m)
,
ùL«
X
d|q
|C(m,d)|
=
Y
p|q
X
d|p
e
|C(m,d)|
≤(
Y
p|q
p-m
2)(
Y
p|(q,m)
2N(p)
v
p
(m)
)
≤2
ω( q)
Y
p|(q,m)
N(p)
v
p
(m)
)
≤2
ω( q)
N(m).
Ïd,·‚
|f(m
1
,···,m
k
)|
≤
X
n
1
,···,n
k
/D
|η
n
1
,···,n
k
||C(m
1
,n
1
)|···|C(m
k
,n
k
)|
≤
X
n
1
,···,n
k
/D
X
l
1
,···,l
k
/D
|(µ
k
∗f)(n
1
l
1
,···,n
k
l
k
)|
N(n
1
l
1
)...N(n
k
l
k
)
|C(m
1
,n
1
)|···|C(m
k
,n
k
)|
=
X
t
1
,···,t
k
/D
|(µ
k
∗f)(t
1
,···,t
k
)|
N(t
1
)...N(t
k
)
X
n
1
l
1
=t
1
n
1
|t
1
|C(m
1
,n
1
)|···
X
n
k
l
k
=t
k
n
k
|t
k
|C(m
k
,n
k
)|
≤
X
t
1
,···,t
k
/D
|(µ
k
∗f)(t
1
,···,t
k
)|
N(t
1
)...N(t
k
)
2
ω(t
1
)
N(m
1
)···2
ω(t
k
)
N(m
k
)
= N(m
1
···m
k
)
X
t
1
,···,t
k
/D
2
ω(t
1
)+···+ω(t
k
)
|(µ
k
∗f)(t
1
,···,t
k
)|
N(t
1
)...N(t
k
)
<∞,
ùy²ýéÂñ5.
ef´¦5¼ê,K(1.21) du
X
m
1
,···,m
k
/D
|(µ
k
∗f)(m
1
,···,m
k
)|
N(m
1
)...N(m
k
)
<∞(2.4)
Ú
X
p/P
X
e
1
,···,e
k
=0
e
1
+···+e
k
≥0
|(µ
k
∗f)(p
e
1
,···,p
e
k
)|
N(p)
e
1
+···+e
k
<∞.(2.5)
DOI:10.12677/pm.2021.114057451nØêÆ
4Ra
Ïd·‚Œ±±e(Ø.
íØ1.b¦5nŽ¼êf:D
k
→C,Ù¥k∈N.b(2.4) ½(2.5) ¤á,@oéu?¿
m
1
,···,m
k
∈D·‚kýéÂñÐmª(1.22),…ÙXꌱ¤:
η
n
1
,···,n
k
=
Y
p/P
X
e
i
≥v
p
(n
i
)
i=0,···,k
(µ
k
∗f)(p
e
1
,···,p
e
k
)
N(p)
e
1
+···+e
k
.
y²ef´¦5¼ê, @oµ
k
∗f•´¦5¼ê, …éuv
p
(n
i
l
i
)≥v
p
(n
i
),i= 0,···,k.·‚
-e
i
= v
p
(n
i
l
i
),Ù¥p∈P, Œ±
η
n
1
,···,n
k
=
Y
p/P
X
l
1
,···,l
k
/D
v
p
(n
i
l
i
)≥v
p
(n
i
)
(µ
k
∗f)

p
v
p
(n
1
l
1
)
,···,p
v
p
(n
k
l
k
)

N(p)
v
p
(n
1
l
1
)+···+v
p
(n
k
l
k
)
=
Y
p/P
X
e
i
≥v
p
(n
i
)
i=0,···,k
(µ
k
∗f)(p
e
1
,···,p
e
k
)
N(p)
e
1
+···+e
k
.
ùÒy²·‚(Ø.
—
a·“Š±9“*•ùÚàX•,3ïÄ?§¥,¦‚Ú·?1õg?Ø,‰·#
Œ•Ï.
ë•©z
[1]Delange,H.(1976)OnRamanujanExpansionsofCertainArithmeticFunctions.ActaArith-
metica,31,259-270.https://doi.org/10.4064/aa-31-3-259-270
[2]Wintner,A.(2017)EratosthenianAverages.WaverlyPress,Baltimore.
[3]Cohen,E.(1959)RepresentationsofEvenFunctions(modr),II.CauchyProducts.Duke
MathematicalJournal,26,165-182.https://doi.org/10.1215/S0012-7094-59-02646-8
[4]Ushiroya,N.(2016)Ramanujan-FourierSeriesofCertainArithmeticFunctionsofTwoVari-
ables.Hardy-RamanujanJournal,39,1-20.
[5]T´oth,L.(2018)RamanujanExpansionsofArithmeticFunctionsofSeveralVariables.The
RamanujanJournal,47,589-603.https://doi.org/10.1007/s11139-017-9944-z
[6]Carlitz,L.(1947) TheSingular forSums ofSquares ofPolynomials. DukeMathematicalJour-
nal,14,1105-1120.https://doi.org/10.1215/S0012-7094-47-01484-1
[7]Zheng,Z.Y.(2018)OnthePolynomialRamanujanSumsoverFiniteFields.TheRamanujan
Journal,46,863-898.https://doi.org/10.1007/s11139-017-9941-2
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[8]àX•.êØ¥eZ)Û¯KïÄ[D]:[a¬Æ Ø©].2²:uHnóŒÆ,2020.
[9]Wang,Y.J.andJi,C.G.(2020)Ramanujan’sSumintheRingofIntegersofanAlgebraic
NumberField.InternationalJournalofNumberTheory,16,65-76.
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