h
Ntatio nandconve ton If X s anysetw deoeth eharat
i n e − ge s − rby N0.I−f x i sa realn um
t
fr acio al p ar x − bxc of x s de n te d b hxi a n dt
he to ra n
or
mn(x
− bxdx e−
e w il be d noted by h
T hes ym b ol γdenotes u0l s on t an . b
r r
f
T he F our i tan sf o m o a 1 - p rio dcf u c on f wi lb e d e noted b f
If f an d g a r e e l u n i o s we us e th e nt a i o n
f g to m a nh
sa m e a f = O g. f l − im x → ∞ f x) /g(x ) = 1w e w r ie f ∼ g .
1. n
ro
d c ton . L t θb e ny n u mb
θnc :n ∈ N} isc alle t h e B eyatt seq uen cede e rm in ed by θ. B at ys
ue c sh a v b e n t h esu bejc t oi nen i ve in ve tiga o ni r ec ny e r so
ac u t f o t − h e i r co n nect o w i h s e m rg o ps (ee f r e xm
t e que tin wec nsi e in th e prs ntp ap er A c asi al resuti np i
nu
m
brter
( ee [2 Th orem 9.9)state t hoatf r
e ch r
tonalθ
th
n
ul
sequen ce B c on a is in f iite y m a n y prm es.E quivl a e tl t hs r s s at tha
f r e a cih r ratona lθ w eh a e
xdp ri m
l − im→ inf τ f loorlef t − parenlef tθnc) = 2,
n
∞
w ere au s u l τ (n) enotest e n u mb ero f d i iso s o f n. I n te p e se
pa
w ec nsi d re t he aev r ag beh a i or o τ(θb nc. Th s d o esn ot a p pea
e
t o hav e b e ni n vsti
g t db ef or
X
per
n ≤ xslash−theta
n≤x
n,∈
Bθ
≤x
n
s h a − ll po ve
re a ti on hip we
thef l l wing :
Th eor e mI
T (θx) ∼ −1T 1
T h eo em I I F
me s u e a n
o
r
a mo t
fore
.)
T
θ;x )
l θ≥
achε
=
e
>0
θ − 1T 1x
wi t
1
hr e p e c to
Le sgue
we have
+O
(
p
The e
w el - k o w s um
rm
y de e n o n
θa nd o n
T ;x a ppear i ng o n h er g h ti nT h eo e ms I an IIi sju tt h
n ≤ e s t i u m a e ound n 3 . o w v e
P
T (one − semicolonx) = x log x + (2γ − 1)x+
r e − le a n i − t nt h e p r e s − e nt cont xt sin e the m rerefined e t − s im at
e s i n − t ro duc
m a inte−r m sth are a b sorb di nour r − e r o − r erm . Th eau h − t o r oesn ot kno
wheth er r o w haet x − t en th e x − e po ne nt 5 7 in the tate men
ofT heore I Icanb
et h a t T h o rem b ec m e s f l s − ef − i t he h y
o
i −d1ropp d: i n partc u ari tf o l ow s f r om Le m m a2. 3 be lw
tha t T (px)∼
p T(1;x doe n o t hold fo r an ypr me p T h ead r m iht w on dr h o wv
w h eth ra ve r ion of T h e o rm I I p os i b y w it h ap oo rerer o r t em,m i
h old wi ht “ali r
ha t his i n ott h e c se Mor p recsey we sh a l pr ov t h elf o l o win g
T
h eo r em
III . Let
g R → R be ost−ie−v , in c e asin−g and
unbo u n
e
T
h−e n
f o r unco un a − t by m
an ynu mb e − r s θ > 1 t hereex
s−i t
ar i − t ra i − rly l − argep o i − t ive x s ch that th relai−t on
(12)
| T (θx
− θ−1 T
1; x) |≤
T (; x
/
g
x)
(
. ;
2 . S om ele m
elem m at
o
.
m ta . I nt h − i ssec i − t n w e
req u − i ed o − f rthe pr ofs
h est−i m a t ivovin g t e − h di vsi o r
Lem
ma
2
)=
1
col e − c t to geth ers om
ofo
f −u n c
X
n ≤ x, n ≡ h(modk)
4]b
sasp e c la c ase o f a fo row nin [ u i mp ct in
r ea h p i bracelefttpm
2p −
h−a
T 1 (1; x )
T (h − commap; x) ∼ braceex
braceleftbt
D
2
−
p
1
ot eh rw i se . Wesh l l r e q − uir esom e r s − e u t fr om
T (; x)
da
i
d q0= ox1 , w h o − e − nso−t
umer o r nth∈N w ee
vextendsingle
t
(po w−e
ap k,e qn
se
pnvextendsinglevextendsingle
vextendsingleθ − qn vextendsingle < vextendsingleθ − q
F o ra n yi n r e a n
f un ct ion g : R → R s ats y
p r in ipa co typ
ε
p − n+
n + 1vextendsingle.
() =1
g
1vextendsingle
vextendsingle
wesa
y t θis o
nine
xmation
apr
l
0,
sta
eas ret ere exists afun c t − i o n g = θR
→
one − plus
g(x =
o
O( g
R su
h−c t h a t
ε
x)
dsu ht h − a t θ i sofprn
i − cp a − l co y − t pe g.
Then x − tre l − utist a n ro m bracketlef t − seven , period − p42.
h e e − a d e w i l − l ob er et hatou
s
definit on of p r − i n ci a lcotype s gh l y o − m e est r − i tive
l
w e a e a su ming g(
X
an
| csc
πn θ| x log
x + xg( x).
1≤n
H
e − re th eimpi−l edco−n s ant d oes o − n t
x
depe d on θ.
T
lessequal − one
<b
s in g m q
b−
1
h − e fol wi ng l em
a sta ken r − f m [, p. 96.
m
Le
m ma
26 . Le
θbe
i
) ≤ ox.T
h e n o r a≥1 e − one − t≥ w e h
a v e h( x
D(θx) ≤ 8x−
0≤
≤h
x
It i w el nko w n(se ef o − r xa m p le 7comma − bracketright C h. I ) t h a
t f o − r a n y i r @o n a l θ t h
num b r s qn sti f y qn+2 ≥ qn + 1 + qn T h e − r f e h eu
n − g : s ) l o gx
−1
Dθ; x L
m
e
xg(
x)lo
e t M b ea na t − u r l − a n u mber .
= m/(M + 1 a d for m 6= 0 s et
a = − π m(1−
e
|m
e
x
Fo re ac h in e g − e r
˜
)ot(πmwide)+
|
m wri e
m|.
2πim
yn . m l s
n
0
e
o
.1
1
ha
h
ooetrpl
M ( )= 1 ≤| m |≤ M a
a
m
y (tt= n hi − 1/ 2. T o x b a p tr c i e w t
l − l owing (s − e e [ 3
)
T bheor−e m A
ch n a ru
u
r M we
h
e h a−v e
1−
| ψM( t
)
− ψ()
≤
(2 M
+
2
(1
˜
−wideme
0≤
W
e nd t ounf o t h
lary
“rr r ” | M
. Fo
(t
t.
m|≤M
−ψ().
e chn t 1aln
M + 1) |
2
b−e
M
h
av e
e
the fo
X
|m
X
2
parenlef t−M
+ 1) −
=2
X
0
0≤M
= 1parenlef t−M
X
+ 1) −
eπ
M
1
− e−
0≤M
≤ 2parenlef t−M +
e
. c f uc o n
o ft h
1) −
nd
−
II F
=2
M + 1)−
h
−t
period − one)
parenlef t − T θx) =
X
(n)θ(n/).
τ
n≤x
w e c n u s F or i re
of h e su ms
m e h d s han d sd e
o f 3.1)b ya l na r
(m /θ, ) = xτ (n)e
π
cm
bn
P
tio n
m
n
Es t m a ti g th se esu ms byL em ma 22 y i d s Th or mI die ct y
ut a mo
n T h o r mI I
d l ca te a n al s s is n ede dt oob tai
d
ragven I ε > l − eξ e −, ξ f i+e i ror
− +
0≤
ξ≤
n
Z
−1
ξ≤ξ
Z
−
1
(3 )2 θ
T
+
−1
− ε3/
≤
b
M M
X
τ
X
n≤
1
X
m|
n≤
≤ Mb+
n
(n)
(m)e
epsilon − slash3 ≤ ( −)and(3.5)that
X
τ n)ξ −b 0)
ε
(34
≤| mlessequal − M
1
ξ
s−summationdisplay−u tin
θo−b
n g(3
πm
X
+(
n
+
≤
−
(3n
n≤x
X
ξb−
(
1 ≤| m |≤ M
− ε)T ( 1x
≥
an
(θ; ) ≤
T
one − semicolon − x
+ T
asa o r e m ips v − eperiod − d
I I fu nct i − o n a s y − f i ng
(1) = 1, g(x = O( l o g 1+
Let θ > 1 bea f i xed irr t i − o n aln umb e r suchtha t
s e q u n e h t h s f a t n h c o jud n t o n wth θLe m mh a ..Ii a
d
e ia
co n We writc e ξ f o =
of
h e o re m
ni mm
ou
θ − + ψ()
(36 ξ(t) =
S ce θ is rrai nal w e h a e n/θ ∈ Z ∪
so n uin g ( 3 6 ) in ( . 1 ) we obtain )
s . 3
(
− ψ(t +
.
(Z − θ− 1 only h en
w
(
)
(
n − lessequal
x
n=0rn
= −1
))
T (; x)
=
θ
−1
θ
θ1T (; x) + R(; x,
37)
nx
say
ee
ed
e nfi equal − e |(
M n − theta
−
ψ
+ S1
S
2
comma − vextendsingle (
)
(
θvextendsinglevextendsingle
o
b u n d f o r| Szero − barand u
fo−r 0 ≤ x
≤ 1), e − w e e t h − a t t e − h t c − oe f f i
i1s
amo − f ψa − s
t−n−e
ify
|
X
( n + j)vextendsinglevextendsingle
n ≤ summationdisplay − x
= vextendsingle
a
meπ jm
θ
(3.9)
X
τ (neπ im/vextendsingle
≤ two − one ≤ m ≤ M mone − minusbar−T (m/,
X X
e h
2e πim/ x
x − slash
l≤
X
n
≤ xl
O
l≤
S i nc
1
s ofP p r nc p
o−t
e−L
Pa
2.5 i m i e − s
cparenlef t−pi lm/θ) |≤
2
l ≤ 1/12m/ 2
12
2
f o−r
a
period − zero
m
(12m/)ε1+
b n i ng (3.9,
(x
1/2M 1/
)
(
3.1
)w
e
(
vextendsingle−2
n≤x
Td en t − est a − r n o
vextendsingle
θT vextendsingle
rm.
We
θ
T
p art it−i n − o
the n a t r l nu m b e
Bin
an
dn


 , n


mn |theta − n 
T 
P
∈ Botherw ie . T hen
(3.13)
S2
θ1vextendsingle≤
vextendsingle
e
τ (n)
l
rv ia ly|
ψM
− 23
+M
P
ψ|
−
≤ 1 weh ave
tau − parenlef tn.
a v e nA
|#( A)
− 2xM
− 2/3 | lg2 + x
soth
#
U ingL
X
3
/.
emm a 21
w − e fin
τ (n − parenright x1 + ε
M −2/3
X
τ
element − B
e
h
(314)
one − two
a n d three − parenlef t14), a n d coos
i
and u ingt hs in ( 3 . ) w esee
ti one to et h − erwt−i hLem
o I n o t h av e T (θ; x t h a θ = θ1 + θ0w he re θ1
d θ0 > 0is sm al . T 0 nw
h−t
q
/θ.T
hav e bθn
h
c=
b1
nc f r0 n ≤
= p/ q i s r a t − io nal an
x, say ,w h r − ex is o m par abl ew
bys h w ing tha a n
m θethod int hss c io ni vag c u oely r e in sen t p−−−o spi e ai ly a re gar s thn e O ap
p ealo D ihl tst t h ere mo n i m si npr gresso
P r oof o f T heo rem I I . Fora n ar i t − rary rea lfun t − c ion G
wewi alla s − eq uen e p1 < p2 < ... o fprimes G− good iffor a − e c hn atur l
νm be−r
we ha ve
≡ 1(modp),
(. 1) p
4
period − two) n+
+2
−
1
pn
≤
n
+1−
1
ad
(43
the
pn +1 ≥
gnym Gi
−
1+
o−o
n
Gpn)
sen
i−t c
e
q u ce
p = pj1 − 1.
es et P0 = 1 and f − o n ≥ 1 we et P n
P a − s tisfie s ( 42 wec an l e − a rlyd efine area nu
X
P
nces
+
Q
=
j ≤ npj. Also if
mbe θP byset ing
t
fo low st ah h her rex s u n ou tab l − y am y − n re la n umbe rs7→ θ >
a
θ = θP f − o r so me G− o od P. n
=0 j q b e d s − t − ict
i1 suc hv t h
an
sh o w
th t θ
P 6= θQI p
1 6= q
1 one c e k se sl y h
bθP c = p1 6=
elyt
q − one = bθQc
os th atc rtin y θ 6= θ
teer su h th t p0m =
6 q0m a n d sua p s posee with ou l os of > gen eraih t
h−a
ip m < q0.c W e ils howth a t θP < θ. W ehave c eart y s
X
m w Q
Q
−θ =
n ≥ m(n−
P
( m+
(qm−
+1 21)p
−
(m 1m1
+
0≥−
p − prime 2
c c musionf o l w s
sar
nec
m
Al s
Bp 0 +
o
0) T he of r − e
Phn n=P>m
−
m
−m 10m..p n0 = Q
w e s hpv−a e
1
q
p
c
eawi
o
+
mte
≤
ym
p
n
2.Henc
0
r
P
3Q
≥
m)
m−
X
≤Q m
1
(q0two − parenright
o
asome
m1
0
a
<Q
0
0
na
d
X
n − greater m n ≥ m+
n1
1
X
2
)n < Q m (q0)2n = Qm
(pm
n1
m ≤2 adt
e−f
ction x
7→
c e
minus − onex − twoi n − i r
ing
a
sth
P n< three − Qmq m.
P a
P
Co m bi g ( 4 ) – 46) w e
hve
W eca nn o w co m et ethe
nc i n G( e − pndi g n − og)uc t at 1 parenright − f s − l
m
ue−n c
w−h e
(m)02.
period − seven)
nd ea − chn
1
θk = pkP k −
one − lessequal
.
p
2
k
= bθ
f our − period7 )i tf
θkn
kn ≤ θ
l−o
n
l−o
ws
nc − P − k − 1 ≤ θn − P k − 1
whi
(.9)
(s − i n
θ−
θk
≥ n− 1P k − 1 ≥
2P k − 1pk = 2kP
X
pj1≤ 1/2 f ore a hc jcomma − parenrighta
48)hod
≤
ndt h − i s
a−c1 la med.
(20braceright − equal
con
(
{θnc :
)
k−
s
sesmh fo u m o {.n : We ≤ b θe / v p h }tt her ]es E−proportionalueFc arrowlef t−
P −1c n1 ree sid u cla
en
it−t
e
t a s s u − arrowsouthwest existential − i − arrowlef t − negationslash0nis
uset of E .
s ttin
g
nd
n
e
n
n
he
rp0) = 2pp 2− 1
an d
r(h ) = p
−
1
f
h 6= 0.
n )
eusi
( g )(f our−period7
pk
pk
eto
≥
r ea −l − aue d
f i − nea
W
N ⇒|T (
G:R
m − pt yfo e c − ah
→ R
rim p
n − cion
−r
;)
G0 o nt h
(
c − h oice of h a nd p).
Gp >
fore ach
≥
pk
W ec nc hoo se
af n c i − t on
2pG0p )
(4.11
n d al oin c − r e a i − s ngrapi d − l y
eo
u h t oe sur eth
a
(4.12 )
f o − r a G− g o d − o a n − d s e t
parenlef t − f our.parenright − eight, (4.1zero − parenright a n dth e de nition of G0(pk) t at
; xparenright−bar
(413
≥
X
r
h) − θ − 1
T (1; x)
=
X
−
|(( k;x ) −θ 1T (1 ) x |
T h, p
−
;x
r
()T (1; x)
¯ pk ¯
(1
≥
2pk
s − e x > (pk )/(pk) a n d g i s i c s − a − e n g , th er
pk .
(|T θx)
−
−
θ1
− 4pk
1
)
1
T ( ; x)
e − l a o n (4.1parenright−two i m
av−e
x)|>
; (1
T
h − us (1.2 f i − a for x = 1/2p0) an d by ( 4 . 2 ) uch an
c hosen a
lar e a s w epla s − e. Th ep ro fi s c o mp t − ee.
m entsV . T h ea uth orn i s n − doand
eted t
sup po t
x
cn
be
pli
nd è s F an c parenright − e, Le
197
o − te Ser.126,
P
c Lon do M at .S oc.(3
38 1(979) 385 – 422.
E. Hl
L
−
ang Int ou c i
Por t
an
[
.H u a I n to d c
nto N u mber T h
ory Spr nger 19 8. [
S
t
to Di o h
tne A p roxi
o, Ad d s on W esey ,1 966. H . A
ma i
K . B . S tola sky , W
tof f p a ir aes m ig r ou p inv
rant
n
o
ah.8 5( 19916
− −82
e
Ram aν jan , o m
f r mu a in t
e an aly [ 1
J D t. Va al , r S m
1eext em alfunci onsinFo uri e an yss , u l . Am er . M
ah. So .
e
o
r
t
S
r
ali
1(19
ART − M ENT
OFP
URE
MA
5)18 36−−21 .
T HE
UN
E
SIT
OF
IL
V
ER
P OL
B − OX14 7
RP OO L,L 693B X,U. K
andinrev
LI