t oduction . Oten , w − h n
cn eringthe n − i tgralov
r
le
i − ngn−et up nthest uc−t ure of th e und e
a
d
b thin n gt he set o ve r w ich h e var bl sl iet on e so sm allrca rin alt
y The m a nr es u t o fht i pa p r ist he fo lwi n g t h ore co nc e ni ngt he m i
no a ce st m at e asoc i at dwi t ht he r r e sent t ono fa
int e er b y egh t c u b s f r m (os e n s ib ly a rb tra ys et .
T
he
ore
m . Su ppo s eth at m ⊆ [, 1],
X
G( α) =
(αn3), i ⊆ 1, P ∩ Z,
ederr om
ter
e
i
an
t
d ha
fo
n ∈ Gi
rs o
]
m
γ,
)
αu∈m G(α
> 0,
Y
ved 1 th rou g ht e e of a a
e le mm a ( Le m m a ).
i
e − e − period a − G n loT gue y sf th ep iege – su t fi
( .
n t o
e m a o r a c s an e r a d yh an led ( th eg n e r ain g f u c i
n s Gi( α) ma
be w ll app o i m te d to o b t an asy mpt tic f orµ a . Ad diti o al y, t he G sh ou l
a
u
bes ffic e nly d ense ( e the minor rc bo nda s n t h e The o r e m w
o verw h m he m aj or a c o ntr uio n ) O fc ou sea Wye l typ e mno ra r ce
su er iort
e
u c h mo re a ts act o
ouldl e aom
o t h on eg ivna bo vw
t im
t
a t e p
r
b
l
f ff e a n d i
p l
W e p r v de w oa p c tions of o ut ech ni ue −−−fi r ti mpr ovng u on th
a
ero tem a s ded c db yV u gh a n [6] i n hs mp ort a n t p ap e ha esta b l he teh as y
t
s
mp o i f or m al o re igh cu be
Coro lla ry 1 . T he
n umber of r−e pr se e nt ai−t on s , r8three − parenleftncomma − parenright
of
nas the
u m of eigh
cu b − e sofp s − o i tive
Γ4
3)8
)
r83(n) = Seight − comma3(n
(/
Γ(8/ 3)
He reS 8, 3n) i sth e usuals n g u l − a rs r es sats f i n 1
ε
era r or o of deh r (l − o g n)4/πeul−2
tha n th e mai n t
ma + s a − ml − e r
shar pr e ult s o f H llan d Tenenb au mf or H ooley s ∆un c io n ( s e [2]).
R oth [4] h a − sproe−v d th at alla r − g ei nte gr−e s a r − e the sum
f − o a cu be a − n d s
c ub es of pr m es .
He d − o esn ot , h owev ecomma−r e − s ta bis an y a
symp o − t ti formu O urm ethod d opr vide a s − y m tt−o ic−r esuls−t , bu f o a ew
enu m ber o − f p c bic s u mm ands .
C
or oll a ry 2 . The
nu
mber of reres ent aio ns R6, (n), o fna sth
e o si xcubes
fp stiv en t ges a n d w p oc ube o
f rimessatis fie s
f
o
oi
i
e
r
t
s
p
R
w
re
h
Cis n
S
n) = q1
a − bso ute o − p s t v ec
onsta nt ,
X
X
∞
q (
a = 1q −
∗
2(, a)6S (q , a)e
φ(q)
−
)
(, aq=1
q) is u l − e − r0
t − oi
Sq, a) =
fo
L
m
a1 D
t
ncti
,and
e
tle c
)=
1
.
R
mmaDu
a n i − t er at veb s − a e f − o r t hetec h i − nqu e introdu ce b yV aughan [ ]
too bt asy mpt ti for mula or e ght ub e − s an de xp oi t d morefu l − ly n
[seven − bracketrightt o t hev erac tyo fth e asympto ticformula f − or 2k k th p owe
r s when k > 3
a
2.No t tio n . A su sua l εde no te a s ffi int ys m all o t eνm ( wh c h
r
cnha ge fro mno e o ccu r nc e t the n e xt , an d d
o ad
v
Vn g
o we l - k o w n n tat on (w he e i m ict co n s tan s ræ f un ct
a
tm
X
F (α) =
P
of ,
3
eαn ),
n1
t e st an d rd g e era tin g fun ct oi n f or c ub es, n d t h a t
X
G(α ) =
e(α3), G ⊆ 1, P ]
F − o r p X = X (% ) = {1 ≤ n ≤ P : p | n i mpi s pelement − negationslash
r − ec ord es r − e l lem m a fo fr u t u r − e e − r e e
Z
0 | F (α|
a B
Lem m
P 2.
dα
( V augh an , [ , Th o rem
Z
0
F
1
2]). W
e hav
e
1
tinso
m
+
E P 7/2
s
m
m
D
S
p
he
s
≤ i1 ≤ P/M, M <
M −9 /parenleft − lgP )5.
m
e
W
re o ss
t
m ⊆[ ]
m
c .
3 4
t
f γ
om
γ
vextendsinglevextendsingle
e(n3)vextendsingle
P o of−period Le t
de r − quoteright s ne q uat−i y ,
Z
8
Bi
= n − elementi
( Z
Y
Y
i − equal
t − ie − n
|G
8
)1 slash − eight
H
m
equa t ons ,
Z
Z
H −i
parenright − alpha| 8dα
≤
Z
H(parenright − bar dα ≤
| F( 8
m
so , by Le m
maB,
Y
.
i
d
m equal − one
8
≤
d
b
i=1 m j=
3
,
m i(α|
whe
w − e e − dfi
b
|G
i
s ffi
j = 3G
8
b
| dα
(α) |
| j ( α | dα +
n∈
∩G i
n−e
×G
D=
{( x y )
, ∈ G1 ×
≤
: (x, y)
≤(o
1G :( xy )
1so t
r y , B C a nP Dpa r i t o n G × G
| 1G (α)| 2 =
e − parenlef tα (x3 − y))
X X X
(∈P 1
)
=
+
+
(x
We
o
e(parenleft − x3−
+
y ) ∈A
wd
∈ B (x)∈
C
Ru
y
tS
(x
o
n
m parenleft − x − comma
o n t he
J ( B) +
t - ha nd
J C)
S
o−f
de
JS
)
j
3
.2)( fo
+ J ().
J parenleft − A)+
i
Z
X
X
3
3
3
vextendsingle8
vextendsingle
d>
an d
(l gP )%
Hö l dr ’ i eq u
b
> ogP %m
lt ,th s is
am
one − elementA
(3. 1)
na
P%
vextendsingle
(d
P
de
j
d( x
=3
ost
vextendsingle(x, parenright − element) A
Us n
J A) P
)y
m parenleft − dx
Y
n
xe d
d − alpha
−
)
y)vextendsingle d α
parenright − element
1
isat
x
mos
+
4
m|
ng t he r a n e o
ealpha − parenlef t3x − y3))vextendsingled α
1/.
B ou n d ng t he nu mb e os o lu to ns ofth e und e r y ng d iop h
Z
X
∈ Ae − parenlef t (x3 − y 3 ))d α
0
jequal − three
n
dntgreater − o
≤ d
.3) J
notJ
Z
3/ γ
P
P
vextendsinglevextendsingle
m y ∈ Be
−
B
X
vextendsinglej = 4
¨ ld quoteright − r si equa ity we fin d
o−
P 34parenleft − logP )γ(
(
X
vextendsingle
t a t J (B)
s
parenleft − alpha
− y3))4(α)|2alpha − d)1
y − parenright
8
Y
)1/8
o z − ie s
g G4(α) b yn F parenright − aft olow v e ma
ep ain
αx − three − y)vextendsingle−vextendsingle−parenright | F (α
Th einte ra−l n − i 3.) s
ualto t − h e nu m
e
3
3
3
3
3
br
th en
u5 mbe
) | d )1
o sou−l t on so f
3
w i t xj ∈ 2G1y j ∈m G1f 1c≤ r js n − lessequal t comma − Pheit n d (xj , j =≤( (
l o g P )%, w1 e − seven r e b (xm mjyaej)= w1. S e t e n i f o mxj n j w e
em j a
summationdisplay − P
f
mα
)
=( ym) 1 = 1( α
y
,
g m ( α) =≤ 1u
P/m eparenleft − alphau).
y
m2 ∈ Q0
m − one − parenlef t
)g
Th
en
m1 α
) m2() minus − parenlef t
m2F
(α
B
Z
(
1
t − yjin
(.)5.
B
q ua t
t
u
1y
t
t
| fm ()gm(αm1 )F (α | dα
2
mcomma − m ∈ Q0
Z
( 1 3 2 )1/2
X Z
=(
( 1| α
3)mw e ded uce
o)zie m ∈a Q 0
m
h
at
) | 2d
(integraltext − one
0
ry i − n g d α
Z
0
0
α) |d parenleft − three.) isb
|F (α) |6 dα ≤
α
)1/
ou ndedb y
Z
|
α4dα
| F( αparenright − bar8dα
1
/2
Usn g Le m mat A a nd B , w e concludet hat
M d , l m ∈ QM 0 = 2m d (α)
d ad i − c bloc s Mt =( M t1, M t]. B y H ö
(
X
Z
| gm(αm3
()2d
X Z
four − bar − f
3/
(
α
4
34
2
1
≤
M
t−1
α
| g m( m
t
)1/4
w
)
= ,
≤
1
parenleft − three.8)h−t at
t− 1
P
P
t
P(
o
gP )γ
(
/5
M
− 1(
−3
/
.
Upo
( 3 . 1 er idJ (C) nalpi e c e , J
of J 5/4three−minus
P − f ive ogP )γ+
Z
e bo on ie rt h e fi
h
b α2
| G(
Y
(31
)| j = 3 j |
X Z
X
.1) J (D) =
−
i
% /ad8
m 1
parenleft − alpha( x3 −
| G(α)dα
| Gj (
y))
m
Afte
e tn d
t h er ang e
X
o−f
ne x ra t on g
( y − three
dα1/4
∩ G1e( α
∈X
(
Y
i
Z
| G(α)8
dα1
− )vextendsingle
(
n d the
a
with
firti
g
nt
e
rai n (13. 2)i s equ l t o th e
@
n ds y t h eXs−a m e 1 an d (xyj) == d. a −y 3 s − ii+ n mo−
a n the va uea t mo vst a n (3 .1) t e o e i 3 1
Z
15four − parenleftBig
vextendsingle
X
o − re
s
th
3vextendsinglevextendsingle 8
)14
n ≤ P/de
αn)
1
P
0
5
By
L e mm a B
t h si s P
d
.Su
mmi n
| (1α
j
|
+ O(
h at
m
o v ed
w
− /4
e co n c u det
fo l − l owing
boun period − d
Lem
X
7b,§ ]).Le
3
n∈ X
T he n
Z
| Fb( ) | 8
0
g − ene
e
Z
I
Q
=
| G(α) .
8
m
8
mG( α
≤j=
s
ffi
|| Gj
rea
e
It
forn − yδ m
Z
a
,
Ij
li
R
j
m
(
|
bG
m j
57(
=
j
m
j
1(α | α
P
Ij m
,
G1
t
Ex te n i n gth e ra n g e o fni e gr at on o nt hefir t ingr e al
m
j(α) | d
weo b
α)8d
0
equation ,
Z
Z
b
1|Gparenleft
− j α)8d α
I f o l−o w r−f m
m
a
y
dhe
Z
3
I1 4
tha t
|
m
m
g
e
1|Fb(α) | 8d−alpha
Le mma E
o
I1
≤
= | G1parenleft − alpha)8dα m
I1
|bG (α |8 dα1/ + P 5ogP parenright − minusδ,
m
r
I1 ea me nt
d
n
r a−d y
p
8
|G
1 (α) dα
b
o i d dw a e f rom L m
e
an d (4.), t eT h eor emi s
period − three
e−d
Z 1
|Fb(α )d
isequa l
α,
to
J
= #{x ∈
X : x3
+3
+3
tin
c ult .
T he au t horgladly
offe s
a pony f o−r a p ro
ha−n th period − X − f − oesi−ra
aspe cal I t
B efo e we a pply t − h eT he orem , we n ed a s uitabl m n − io ra
r − c e t − sm
The f − o llowi g resu t ofV a ugha nis s ufficie t or t he ap plca sh llcon si de−r .
L m m a{ F(V vextendsinglea ug a n, [, Le mm
1) .Le t
a
vextendsinglevextendsingle
m=
α
α: vextendsingle −vextendsinglevextendsingle
Thenu n f o−r mly
o−f r
α∈
()
1
a
3/4}
m,
P/
loP ) /
ε.
b o undsh ofHa lla n T en
n t heTheore m C s rto ettem O l ce−q a l to rF (α ). T h ei m n − iorarcins te
crr sp ond so r i (n), the n um bl e r of rep esenaton s f n a sth a n yre
a − l α w ecan find i n g − eer s a and q wit h (a, q) = 1 and 1 ≤ q e
≤P
w e−r e
vextendsinglevextendsinglevextendsingleα − a
vextendsingle
vline ≤
α
line − one
.
vextendsingle − vlinevextendsinglevextendsingle
1
the ∗ s gi i fi − e s t − propersubset
[
t ot work o n V a nd P n m =V \
t an d rd te c h que
hth
euonni is t a
h
1
o re a
1≤a≤
]
M We
m ay
eat th m ao r a c s
g
Mu s n
B
conditi on s ofLmt mp a s F s o , δbyt
h
Z
Z
)
F (α)e(−nα) dα ≤
dα
P 5(log P
)
bar−F −parenlef t−alpha
.
T − h i co mple es th ep oo of he corol−l a r − yP
P ro o f o
f C or o la r − y period − two L − e t
e − fi nedi nt e − h proofof C o oll
∈V
vextendsingleα − a
qvextendsingle
) = 1. Th mN aq
6( 1)
Rsix − comma2 (n) =
c
≤ P q3
n
s
T − ont dna N
eij
,
N=
Na)
M.etn = M\
Z
1
G(α)2F (parenright − alphae( minus − n )dα
0
N
m
n
Z
,
|
α
2 parenleft − alpha)6d
P 5(
o
)ε − 3.
( F
Z
2
(
8
)1
4(
Z
)
n
o − n s deri n g the
u n dr−e y5 n d io−p ha
ds fi s − comma
n t nee t oa a o n , we
n e
fi n d
tn
|G (α
|F (α | 6dα
P )ε− 3,
P 5og
a−n d ( 6 . 2 )
(
N
m t oo
bt a t h e m ai n t er m
β)
=
u − /3(lo − gparenright − u −1e(u),
vparenleft − beta) = 1
P
31 lessequal − u ≤ P 3
and S
(a) a n d ∗q, a) b easde fined in
∗(α
q
1S ∗ (a)v
= φ(q
α
he
− a)
( a)
v
F(
f om w
)−
q
q/
V (α, qa R
ow s t − h a (s e[ 5,
h c hit fol
G (α )F (α)6e(−n α
N
Z
X
m
x−p
>
. F o l o wn g R o t h[ 4,
L m m√ a , f or α
G
lg
i−v e
c o ta n t c1 ( a re
utl
hr
w c−i
s
G(α
f
√
N
√
)2 − V ∗ (αq, a)2 P e1l − ogP(
∈N
Z
K(n) summationdisplay − W
q − summationdisplay
e
q
By
a i−r−e
a−c
a = 0 q)
K(n)lessmuch − n − f ive/
o
q − 6φ(qS − 2
)
√
−1
(, aparenright−six S ∗ (q ,a )2
q=1 (a−comma qequal − one1
multiply − e
h a fi
vline ) W qP
v∗ (β
y(w
ep
n−t r − etwo − bracketright − parenright
i
q tonfi n i t y a d th e r tad ar d ∗t m ates
fo rrwa d
ac
cl lu
2
β)6(
t e xte nd th e
sumo
n
f i−periodn fofC
i−t
or h
e
funct o nv β) a
nd s t ai ht
ati
inv
lingv (β).
C roll ar y 2 w as b enefitte dby a co nv−e rs tio n w
T . D . W o o − l ey .
It m a y b e
1 ( s e bracketleft − five, Ch pt r 2parenright−bracketright . Ase p r − t
si t hefield w lr l e , ad l − e icatet e − at m
T his
prof
of
of t − he
asso
su m of th ree cu be sa n d the c u e o ap rim el ea ds to me ans q a r e re
e
pr ov ng tha t t h ex pe ct d as ypm o t c f or m ula o r h e re pr e s
en t t io n th su mo ft h eec ub e a nd t hec ube o f a pr me hol sa l m ot al
w ys (in s en eo f n at ura de n ity
R ef r nc s
mD
r − d − −yLitl ewood M
L
o
s c m allr expone nn s , II , Mat he mti ka
J
— , ,On
Wa ing s s roble m moo
133( 196) − −,6 – 8 ]
— An e w ite a t ve p m thod f in [
ng0 p ro em, A c a M a .16 (198),1−− 7
7]
M AT HEM AT I C S
,
r
i
e
W
ar
DE PA RM EN TOF
V AND
CEN T E
N ESSEE
37401
U SA
12ST EE N SON
N AHV LTE, E N