The Autocorrelation
Functions
and Joint Distribution
of the Sequences
~/2f,
j—-(/ + T)2f
By David L. Jagerman
1. Introduction. The present day extensive use of Monte-Carlo procedures
necessitates the careful investigation of methods for the generation of random
numbers. In its simplest form, the underlying principle of many Monte-Carlo
procedures finds its expression in the following theorem [1].
Theorem. Let (x,)™ be a sequence equidistributed over (0, 1) and let f(x)
function Riemann integrable on (0, 1), then
E/(Xy)~iV
3=1
be a
[ f(x)dx.
Jo
Thus, the theorem states that sample averages approximate the value of an integral.
It is to be noted that the only property of the sequence (xj)i°° employed is its
equidistribution.
In applications, one may employ several equidistributed
sequences simultaneously; accordingly, new requirements may arise. The sequences may be employed, for example, as bases for decision, in which case it may be required that
they be independent. Thus depending on the Monte-Carlo problem considered,
equidistributed sequences may be required to possess other random number characteristics. Let the sequences (x3)i°°, (xj+r)i°° be designated respectively by x, x(t),
in which t is a non-zero integer, then an additional desirable characteristic is the
statistical independence of x, xM. A sequence exhibiting such characteristics is
({aj })ix in which a is an irrational number [2]. The symbol jx} is employed to
designate the fractional part of x.
However, from the viewpoint of the practical utilization of the sequence suggested above by means of a digital computer, it is necessary to replace a by a
rational number, and hence, to lose some of the precision with which the characteristics discussed above are satisfied. Accordingly, the sequences which will be studied
are
The integers a, m are taken relatively prime. Of particular interest will be the
deviation of the characteristics of these sequences from the ideal random number
characteristics.
Let p(x) be given by p(x) = \ — {x}, then the autocorrelation function 4/(T)
of a sequence (a;y)i°°is defined by
1 N
4>(t) = lim —52 p(xj)p(xj+r).
n-*x Jy y—i
Received March 1, 1963, revised September 5, 1963.
211
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212
DAVID L. JAGERMAN
For the sequence to be studied, this takes the form
m ¡To
\m
J
\m
/
It will be shown that, uniformly in t for the range
1 Ú r < ■y/m,
the autocorrelation
function is small; that is, quantitatively
| *(T)| < m_1/i[.81(2 + V2)"M
provided
In2 m + 33(4 + 2y/2)'lm) In m],
(a, m) = 1 and m ^ 36. The function
v(m) is the number
prime divisors of m. Thus, the sequence ( <— /> Jo™-1is approximately
Let Ha(x)
be given by Ha(x)
= a + p(x)
of distinct
uncorrelated.
— p(x — a), that is, in the initial
period,
Ha(x)
then the joint distribution
= 1,
0 ^ X < a,
= 0,
« á i<
1,
function G(a, ß) of the sequences x, x(r> is given by
1 *
G(a, ß) = lim — 51 Ha(xj)Hß(xJ-+T).
A--.M iV j=l
For the sequence I s—J f lo™ , this takes the form
({^}>«,«í
G(a,ß) = 1-flHa(^j2)Hß(a-(j
+
It will be shown that, uniformly in t for the same range stated above, and under
the same conditions on a, m,
G(a, ß) - aß I < m~1/2[3.24(2+ ^/2)v(m) In2 m + 392(4+
2 v/2)"<m,ln '«
Thus, the sequences (< — j2> jom \ ( s — (j + r)2> j0m x are approximately
independ-
ently equidistributed over (0, 1).
The above enumerated properties
of the se-
quences ( \—j[
show the possible applicability
Jo™-1,(s— (j + r)2> jom_1as random numbers in Monte-Carlo
pro-
cedures. However, an important question is the behavior, from the viewpoint of
random number characteristics, of consecutive portions of the complete sequences.
This is being studied by the author, and an investigation of the question will
appear in another paper.
2. Analytical Discussion.
ment of several lemmas.
The proofs of the main theorems require the establish-
Lemma 1.1 ^ 1
, s
=> p(x) =
-^
¿,
íáAgi
sin 2irhx .
. /,
1
\
—r— + V mm I 1,
.. . )
irh
\ 2iri||x||/
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AUTOCORRELATION AND JOINT DISTRIBUTION OF SEQUENCES
213
where
|ij| < 1
and in which \\ x \\ denotes the distance from x to the nearest integer. It is understood
that when x is an integer, the estimate 1 is used.
Proof. The Fourier series for p(x) is
2irhx
p(x) = 51sin irh
(2.1)
hence, it is necessary to establish
- sin %rhx
The following standard
< min
V'2rfMI/
theorem derived from Abel's transformation
of series will
be used :
ai i è 0,
52 b,
ú B
p=M
(2.3)
CO
51 e-ib
S aMB.
Also standard is the following estimate:
i
52 sin2wpx „
(2.4)
p= M
Applying the inequalities of Equations
1
2||*||"
(2.3) and (2.4), one has
sin 2irhx
(2.5)
h>t
If 2-wt|| x || > l,then
2t([í] + 1)|| x ||
rrti
Equation
<
2ttí ||x
(2.2) has been established.
Consider now the case
2ir<|| x || g 1, then
y^ sin 2irhx _ y> sin 2-rrhx
h>t
irh
a=i
irh
(2.6)
y^ sin 2-whx
i<h<i
irh
Thus
sin 2irhx
h>t
h
-w,
- 2 T
E
l</><¡
(2.7)
sin 2«-Äx < 1
irh
y>
l</l<i
| sin 2-wh|| X
7r//
^5 + 2||x||i^I
+ I< 1.
Equation (2.2) is now established. The Fourier series for p(x) does not equal p(x)
when x is an integer, however, with the understanding stated in the lemma, the
lemma remains correct also in this case.
Lemma 2.
sin 2-whx
<
-wh
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214
DAVID L. JAGERMAN
Proof. One has
y^
(2.8)
sin 2-whx _ y« sin 2xfcc _ y^ sin 2-whx
\<h<t
irh
h~i
-wh
h>t
-wh
Hence
2-whx
52 sin -wh
(2.9)
iáfc¿¡
y> sin 2-whx
2T
h>t
irh
4
The inequality of Lemma 1 was used.
Lemma 3. t S: 1
p(x)p(y)
= l£h<t
52 l<l<t52 sin 2-whx■sin
+
2-wly
n min ( 1, —-¡i—n ) + min ( 1,
L
\
2xí||x||/
.
. )
V 2xí||w||/
where
1JI < 1.
Proof. Use of Lemma 1 yields
p(x)p(y)
= 52 52
sin 2-whx■sin 2-wly
-w2hl
■ ( -,
1 \ .
v^
v^ sin 2-whx
-r—
• ln I X' 9 ,11 „II ) + 1 ¿i<ä<(
-wh
\ zxt || w||/
ísiáí
tr,ifs\
(2.10)
+ ?; ¿_
sin 2-wly
-7—«
■min ( 1, -—n—n ) + V min ( 1, -—n—n- ) min ( 1, -—n—¡
V 2xí||x||/
V ' 2xí||x||/
V 2xí||w
Lemma 2 allows one to write
, \ , s
t^
p(x)P(y)
= E
láASÍ
n^ sin 2x/ix-sin
E
-ñj-
lgjiái
2xZw
x hl
-i-3
^^t,—n ) + min ( 1,
+ 2" min ( 1,"2xí||x
2xí||»ll/J
(2.11)
+ ri min I 1,
1 ^inun• d1, -—T7-i
i,—n-
2x< IIx\\J
V 2x<||
Observing that
(2.12)
min ( 1
■n—rrI mm I 1, -—n—iï
'2-wt \\x\\J
\'2irt\\y\\J
< mm I 1, -—n—¡r )
-
\'2xi||x||y
< min ( 1, -—n—¡ï 1 + min ( 1, -—n—n ),
V 2x¿|| x \\J T
V 2xi || y \\) '
the lemma follows.
Let f(j),
(2.13)
g(j) be given functions of the integral variable,/.
e(x) = e
,
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Define e(x) by
AUTOCORRELATION AND JOINT DISTRIBUTION OF SEQUENCES
215
Pby
R= a<j52g bP(f(j))p(g(j)),
(2.14)
Sh.i by
&u = E e(hf(j) + lg(j)),
(2.15)
a<j¿b
and Sf by
Sf = E min ( 1,
.
, ).
«<¿sí
\ 2xí Hf(j) ||/
(2.16)
then
Lemma 4. t > 1
=Ha|<¿E
E %^
+ ¿¿ir is^á'
S la's'E %*
+ §«,+
jk.
¿x i¿h¿t i<i¿t
ht
hl
¿
¿
Proof. Observing that
(2.17)
sin 2whx sin 2wly = 5 eos 2x(/ix — ly) — | eos 2w(hx + Zw),
one obtains from Lemma 3
/ n , n
1
\->
v^ cos 2-w(hx
Áx)p(y)
= —i
E
E -^-— hl
¿x¿ ígAgíisisf
(2.18)
cos2x(/ix
+ Zw)
-W-+
5
2"
— ly)
1 v^
v^
-s-ï
E E
¿ir íg/iáíiáigí
min ( 1, -—n—n I + min ( 1,-n—ñ
V'2xZ||x||y^
V'2xZ||w||/
)
Replacing x by f(j), y by g(j), and summing with respect to y, Equation 2.18 yields
ß = ^¿X lgíigí
E
52 Ihl a<jSb
52 coS2ir(hf(j)
-lg(j))
lg/SÍ
(2.19)
- ¿
E
E
¿ E cos2r(VO")+ foO"))+ |n(5/ + -S,).
Thus
52 cos2x(Ä/(j) -Zff(j))
(2.20)
+ -2x2 i<*<¡
52 îgisi
52 ÄZ
-
52 cos2x(/i/(i)
+iflf(j))
o<y ¿ b
Since
| E cos2x(Ä/(i) -&(j))|
a<j S 6
(2.21)
| 52 cos2ir(hf(j) +lg(j))\S
á |&,,-i|
I-SmI,
the lemma follows.
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+ ö £/ + ?; S„•
216
DAVID L. JAGERMAN
For the sequence given by
O'f
(2.22)
(a,m)
it is necessary to estimate
(2.23)
R = 52p (~f
j_o
- «) P (- (j + r)2 - ß) .,
\m
/
\m
/
in which a, ß satisfy 0 á a < 1, 0 | jî < 1. Let
(2.24)
/(;)=-/-«, m
(2.25)
g(j)
=-(;
m
+
r)2
-
then,
Lemma 5. (a, m) = 1,
»S/>,¡l^ s/2m(h + Z,m)
Proof. One has
Sh.i\ =\51e(^hj2
(2.26)
y=o
\m
+ ^l(j
+ T)
m
Let
d = (h, I, m),
h = h/d,
(2.27)
I' = Ifd,
m = m/d,
then
m— 1
/
•sp
(ah
i-n
\m
7/
,'
2 , al
2^ el -yj
, . .
x2
4- — (j + T)
m'
(2.28)
(ah' .2 . aZ' , .
.
= <Z"W
2^ el—rj
+ — (j + T
j-o
\m
m
Let
(2.29)
i-o
•fêi'+Î O"+
*>').
\TO
m
then
m' —l
(2.30)
S'¡ =
i-o
/
. 2aZt A
\m'
+ I )j H-r-j)
m
/
One has
771—1
(2.:3D
m —1
/
|S'|2= E Ze(¿(h'
i_o
,-_o
\m
O
7'
+ f)(f-k2)+™lU-k)).
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m
\
/
AUTOCORRELATION
AND JOINT DISTRIBUTION
217
OF SEQUENCES
and
m —1 m —1+fc
S'|2=E
(2.32)
/
O 7'
\
52 e(^,(h' +l')(f-k2)+^(j-k)).
k=o
j~k
\m
m
/
Let
(2.33)
j = k + v,
in which v is a new summation
variable, then
(2.34) IS' I2= ™E"E e (2A (h' + Í)vh) e (± (h' + l')v2+ ^1,),
v=o *=o
yn
/
\m'
m!
)
and hence,
s'fá
(2.35)
E
52e(2a¡(h' + l')vk)
v=o
k-o
\m
/
Let
S = (ft +Z,m),
(2.36)
6=
J
0
,
m
=
m
S '
then
( 2.37)
"E e (2A (ft' + l')Vk) = "E e (?? ,k) = 8 J? e (*£ ,*) .
*=o
\m'
/
i-o
\ra"
/
*_o
\m"
/
Thus one obtains
s'\2^b
(2.38)
By direct summation,
/Sao .
2^ e I —- vk
52 Vk-o
\m"
one has
m"-l
\
yT1
E
(2.39)
h
*_o
A
e ((2ab
—- vk
) = n0,
m" \ 2abv
= m
m" I 2abv.
\m"
/
Since
(2.40)
(ab,m") = 1,
m" I 2abv at most 25 times and hence,
(2.41)
\S'\2 £ 2b2m" = 25m'.
Equations (2.28), (2.29), and (2.41) now yield
(2.42)
\Sh,i\ ^ dV2b~m' = \/2ôd2m' = y/2bdm = y/2m(h + I, m).
Lemma 5 yields a trivial estimate when applied to £/,,_/,. It will be important
to determine an accurate estimate for this quantity.
Lemma 6. l^r<-^,l^ft^Z,
m > 2t
Sh,-h — 0.
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218
DAVID L. JAGERMAN
Proof. One has
(2.43)
I Sh,-h |
E«(-
(i2-(i
+
Since (a, m) = 1, the sum in Equation
r)2)
\
Ç;1 /2aftr A
2.43 is zero when m \ 2hr. One has, for
1 g: T < ^-,1 S h ^ t,m>
2t,
(2.44)
2 á 2k S 2ít < m.
Thus, m 12ftr and consequently, Sh.-h = 0.
m
Lemma 7. Z è 1, m > 2í, 1 g r < —
ZI
^|P|<^¿2V2x
52 isísí
E ¿V(*TT1S)
ft*
ISkS'
+ -^i
V2x
52 E T7
V(ft - z,m) + -¿ a/ + -¿ sg.
ftt
ISAS! ISiáí
Proof. The lemma follows immediately
Lemma 8. For t è 3, one ftas
^Ä
V2x
from Lemmas 4, 5, and 6.
52 52 ÍV(h + l,m) <^(7.101n2Z + 5.261n^lnm)(2+ v/2),'<m,,
ISASÍ
ISiSÍ
ftt
TT
m which v(m) denotes the number of distinct prime divisors of m.
Proof. In abbreviation, define S as follows.
(2.45)
S = V—j=-2
52 52 -nV(h + I,m).
2x i¿h¿t \<¡i¿t ni
Let ft, Zbe restricted so that (ft + l,m) = d, then
(2.46)
s = ^52Vd52
v2x
al™
lá^á'
52 ¿.
iá'S< ftt
(ll-(-!,77l)=d
The set of integers ft, Z for which (ft + Z, m) = d is included in the set for which
d | ft + Z,hence
(2.47)
s^^EVrf
\/2x
<i|7n
E
E ¿.
l<Ag(lS¡áíftt
(¡i*+í
All integers fall into d residue classes modulo d. If r denotes one of the integers,
0, 1, 2, • • -, d — 1, then when ft belongs to the residue class represented by r, I
belongs to the class in which d — r is member. Let Sr denote the following sum
(2.48)
Sr= 52 Ei,
Asr(modd)
l =d—r(raodd)
and S'(d) denote
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AUTOCORRELATION AND JOINT DISTRIBUTION
(2.49)
S'(d)= lSh^t
E
OF SEQUENCES
219
52 L
ftt
lgiáí
d\h+l
then
sú ^
(2.50)
EVds'(d),
^
dlm
s'(d) = 52sr.
r-0
From the symmetry of the sum in Equation
(2.51)
(2.48), it follows that
&=&_,
and hence
(2.52)
S'(d) g S0+ 2 E
Sr.
lSrgd/2
Setting ft = cd and Z = ed in which c à 1, e à 1 are new independent
variables, one has
(2.53)
summation
&-Ud \i¿c¿t/dE -Y
* ¿d2 (\igc£(E -Yc)
c)
Since, for t è 3,
--^
(2.54)
E
1
IScSiC
f' dr
- < 1 + / — = 1 + In Z < 2 In Z,
Jl
X
one obtains
(2.55)
-So < 4
d?
For r satisfying 1 ^ r g -, let ft = cd + r and I = ed — r, then ft ^ 1 implies c ^ 0,
I ^ 1 implies e ^ 1, and
(2.56)
Srâ ,<„<<=!
E cd
tA—•
E
+ r 1<,<f±red-r'
-
-
d
-
-
d
One has
(2.57)
E
-J-<-1+f¥-A-=-1+^-^-+1-^.
~~ d
Also, one has
(2.58) iSe<i±r
E V—=
ed - r
2:
-rx^-
0S6<<±r_i ed + d -
d
r
^d
d
in which the inequality r ^ - was used ; further,
2¡
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Et±z_1 e-4t.
+ i
d
220
DAVID L. JAGERMAN
52<+r_1e
(2.59)
-
-
1 1^2
í±-r-i
+
dx
x
+
j < 2.7 + ln t < 3.7 ln t.
d
Thus, one has
3.7 lnZ
E
(2.60)
1¿e¿i±red
-
r
d
Equations 2.56, 2.57, and 2.60 yield
„
(2.61)
Or
3.71nZ . 3.7 In'f
-;—
+
-™-•
rd
<
Equation 2.52 now takes the form
„/,,,
41n2i , 3.7 In Z . 7.41n¿ ^
o W < -—--1--,—
d2
d
(2.62)
-i--z—
d
1
2^ ,^
.¿r
1<
- r<- -2
Since
4 In2 Z ^4 In2 Z
(2.63)
d2
-
d
'
and
Z Ul+
(2.64)
f^=l
+ ln^<.31 + lnm,
in which the inequalities In 2 > .69 and d S m were used, one obtains
S'(d) <
(2.65)
+
Thus, Equation (2.50) yields
S < -^S. (10 In2t + 7.4ln Z-lnm) E -i-
(2.66)
V2-W
Let 6(a) be a multiplicative
d\mVd
function of the integral variable a, and let
(2.67)
a = Vlai-pr---pf"
be the canonical factorization
of a, then the following identity holds [3].
(2.68)
52 0(d) = II [1 + e(p) + d(p2)+ ■■■+ d(pa)].
d\a
p]a
Employing Equation (2.68) with 8(d) == 1/Vd, a = m, one obtains
(2.69)
d\m vd
Plm
1+J-+
Vp
1
(Vp)2
+
1
(VpY.
and hence
(2.70) 52~ < n
_L + _L_2 +
Vp (Vp)2
•1
-^
j - n»i™
Vp —i
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AUTOCORRELATION AND JOINT DISTRIBUTION
Since p S: 2, one has \/p/(-\/p
(2.71)
OF SEQUENCES
221
— 1) Ú 2 + V2, and hence,
E4= = (2 +V2)'(m)
<*!»>
Vd
Equations 2.66 and 2.71 now yield
(2.72)
S < ^3-2 (10 ln2 t + 7.4 ln Z-ln m) (2 + V2)"(m).
V2x
Finally, use of the inequality 1/V2
Lemma 9. For t ^ 3, one has
-\/2x
< .71 yields the result of the lemma.
52 îs'â'52 t,ft' V(ft - z,m)
is»si
Mí
< ^
(8.52 In Z+ 2.84 ln Z-lnm)(2 + V2)'(m>.
x¿
Proof. In abbreviation, define /S by
(2.73)
S= -^
\/2x
52 iâ'â<
52 ft'\ V(ft - Z,m).
íá/iá'
As in the proof of the preceding lemma, one may write
(2.74)
=VlíS^iSíiísíW
(h—l,m)=d
h^l
Define fir(d) by
(2.75)
íg/igí is¡áí ftZ
dl^-i
hj¿l
then, since the class of integers satisfying (ft — Z, m) = d is included in the class
d | ft — Z,one has
(2.76)
S g y^-252VdS'(d).
V2x
<i|i»
When ft belongs to the residue class represented
residue class. Let Sr denote the following sum.
(2.77)
Sr =
2-1
by r, then Z belongs to the same
/ ■ 77 .
íáfcgí ig/.gi ftZ
ft=r(mod
£=r(mod
d)
d)
then
d-l
(2.78)
S'(d) = 52Sr.
r=0
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222
DAVID L. JAGERMAN
Setting ft = cd, I = ed, the inequalities l^ft^Z,
1 ^ e g t/d, and
«
\igcgí/d
l^Z^Z
c/
d2
For 1 g r < d, let ft = cd + r, Z = ed + r, then l^ft^Z,
c á —t— ,0|e^
—-j—, and
(2-80)
* = „£ts
0^(cd
d
imply 1 ^ c ^ t/d,
l^Z^Z
imply O á
+ r)(ed + ry
d
Due to the symmetry of the sum in Equation 2.80
(2.81)
"'^ÏTÏ
— — d
— — d
Equation 2.57 yields
(2-82)
2-,_,™nT7<r
0¿c¿i=zcd
+ r
r
+
d '
d
further
(2 83)
v
;
E
7,
iSe<ir-i ed + r
~ ~
< -, E
di¿e¿t
- < -j (1 + Ini) < —j-,
e
d
d
d
and hence,
(2.84)
&<i£?+
*£-'.
rd
d2
Since 1 :£ d :S ?rcand
d-\ 1
(2.85)
E - < 1 + In m,
■
r=l r
Equations 2.78 and 2.84 yield
(2.86)
g-(d)<l^LÍ
+ 41n¿;lnm.
d
d
From Equations 2.76 and 2.86, one now has
(2.87)
S < -^S-2 (12 1n2Z+ 4 ln Mn m) EA=V27T
d\m vd
Equation 2.71 and the inequality 1/V2
< .71 yield the result of the lemma.
Let Ar(m) designate the number of solutions of the congruence
(2.88)
af = r(mod m)
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AUTOCORRELATION AND JOINT DISTRIBUTION OF SEQUENCES
223
then Lemma 10 provides an estimate for Ar(m) which is needed in the estimation
of the sums S/ , Sg.
Lemma 10. (a, m) = 1
=» Ar(m) = 2'im)+1V(r7mj.
Proof. Let A designate the number of solutions of the congruence
(2.89)
/ ■ „(mod m),
and let
(2.90)
m = 2"P1ai---Pkak
be the canonical factorization
(2.91)
of m with
a = 0,
ai > 0,
1 = Z = Ä;
and the Pi odd primes. Then by the Chinese remainder theorem,
k
(2.92)
A = T]jTi
i=i
in which Ti is the number of solutions of/ = „(mod p"1 ), and T is the number of
solutions of j = „(mod 2"). Consider the congruence
(2.93)
/ = (r(mod pß).
If pß | o-then the congruence has one solution i.e. y = 0(mod pß). Accordingly, let
(2.94)
(<r, pß) = p2y+s,
0 |
ä g 1,
in which it is now supposed that 2y + 5 < ß. Divide Equation 2.93 by p2y, then
(2.95)
j'2 s „'(mod pß~2y)
in which
(2.96)
,■= pV,
o-= p2V.
One must have 5 = 0; otherwise, let 5 = 1 then „ = pq,p \ q, ß — 2y = 2, and
(2.97)
j'2= pq(mod pß~2y).
Sin p | pq, p | pß~2y,one has p | j'2; hence p2| /2. Also, since p2| pß~2yit follows that
p2| pq which is impossible. Thus 5 = 0 and („'', p) = 1. Let jó be a solution of
Equation 2.95 then
(2.98)
where t is an arbitrary
(2.99)
(jó
integer.
+ pß-2yt)2 -
Multiplying
<r'(mod pß~2y)
Equation
2.98 by p2y, one obtains
(jo + P6~ytf - a(mod ps)
in which
(2.100)
jo
= p7Jo',
and
(2.101)
jó = o-(modpß).
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224
DAVID L. JAGERMAN
Since for all Zsatisfying 0 ^ Z < pT, one obtains solutions of Equation 2.93, there
are p7 solutions of Equation 2.93 for each solution of Equation 2.95. Since („ , p) =
1, the number of solutions of Equation 2.95 does not exceed 2 when p is odd, and
does not exceed 4 when p = 2 [3]. Thus
(2.102)
From Equation
Tl = 2p7,
ß = at,
T g 4-2T,
ß = a.
2.94, one has
(2.103)
p7 = V(^p^);
thus, Equation
2.102 may be written
Tt = 2V(<r,pi"0,
T g 4V(^~2°).
(2.104)
Let m be odd, then
(2.105)
A = If Ti < 2" A/fi ((7;pj-«)= 2"(mV(^).
í=1
r
¡=i
If to is even, then
(2.106) A = TU Ti < 2k+2i/{(T; 2") ]J („, Pl<") = V^+W^
,=i
y
i=i
to).
Thus in all cases
(2.107)
A g 2Hm)+W(o^m).
Since (a, m) = 1, one has
(2.108)
(„, m) = (r, to)
and hence the estimate of the lemma.
Lemma 11. (a, to) = 1, 0 á « < 1, 0 á ß < 1, m = 2wt
^Sf<
8(4 + 2V2)Km)
[Vto + ^P],
S„ < 8(4 + 2V2)Km>
[Vto + 2^1.
Proof. Consider, in the sum
m—l
(2.109)
I
1
Sf = 52 mm /1>
j=0
\
2-wt
regrouping the terms so that all j for which
(2.110)
constitute
(2.111)
aj = r(mod to),
0 = r < to
a group indexed by r, then
fi/ = E Ar(m) min / 1,
2xZ
?«
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AUTOCORRELATION AND JOINT DISTRIBUTION OF SEQUENCES
225
The estimate of Lemma 10 yields
Sf = 2Km)+1
52 VÖW) min /1,-T-1
(2.112)
r-o
„
Let r be restricted so that (r, to)
(2.113)
= d, then
fi, = 2'(m)+1
E Vd E
d|m
The set of integers r satisfying (r, m)
S, = 2^>+1EVd
\
¿XÍ
m
= d is included in the set d | r, hence
min /l,-*
0Sr<m
rf| r
Introducing
r
— — a
„
E
d|m
1
min /l, —
0gr<7,i
(r,>»)=d
(2.114)
. \ r
2-wt
I
-
\
¿Xt
.H r
— — a
TO
the summation variable c by r = dc and setting d' = m/d, one has
(2.115) Sf =22"(m,+1
SVrf
dl »n
S
min il, -
0<c<d'
I
d'~a
Let
S = 52 min/ 1,
(2.116)
0ác<d'
c
2-wt d'~a
Then the sum S will be estimated by consideration
of four cases.
c
1
Case 1. —1 < —,— a = —-.
d
—
One has
c
(2.117)
E
-i+i-.
min /1,
0SeSi'(ii-:J)
2xz(l+|-a)
d'(a-i)
I dx,
min / 1,
Jo
(2.118)
2xz(l + d?_B),
< 1 + d' / min ( 1,-)
Jo
< 1+
Case 2. -i
2
\
d ln nl
Vi"
< ^ - a = 0.
d
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2-wtu/
du,
226
DAVID L. JAGERMAN
One has
(2.119)
d'
= a~d"
and
< 1
52 min/1,
0< c< d' a
1_.<x«-d,
2xZ
(-i),
(2.120)
i eux.
min / 1,
< 1
d' In 2xZ
< 1+
Case 3.
xZ
'
c
1
0 < -, — a ^ ct
Z
One has
c
d'~a
(2.121)
= d'~a>
and
52 min [ 1,
d' a<c<d'
I dx,
min / 1,
2xZ ~ -
(2.122)
2xZ
= l+d'(min(l,¿)
min ( 1,
< 1+
Case 4.
1
| du,
d' In 2xZ
■wl
C
- < — — a < 1.
2
d
One has
(2.123)
o7_a
E
d'(a+i)<c<d'
1 - d> + "'
min /1,
2xZ (l -
« + a))
^ 1+
min / 1,
•'d'(a+5)
1+d'l
< 1+
l dx,
2xí
(2.124)
min (l, ¿_) du,
d In xZ
xZ-'
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I +•),
1 - ^
AUTOCORRELATION AND JOINT DISTRIBUTION
227
OF SEQUENCES
Thus combining the estimates obtained in the above four cases, one has
d! In 2xZ
S < 4 +4
(2.125)
wt
and hence
sf < s.2Hm)
52 Vd 1 +
(2.126)
m In to~]
■wtd \
d\ m
where use was made of the inequality to = 2xZ. Equation 2.68 permits the following
transformation
of the sum 52d\m Vd
(2.127)
EVd = njl
d\m
+ Vp + Vp2+---+
Vp"]
p\m
in which the product is over the prime divisors p of to. Since
1 + Vp + Vp2 + • •• + Vp" = —P~-
Vp - i
(2.128)
< Vp" /P , = (2 + V2) Vpa,
Vp - 1
one obtains
E Vd < (2 + V2r(m,Vro~.
(2.129)
d|m
Thus
o
^ or a i o
/K\Km)
5/ < 8(4 + 2 V2)
(2.130)
/—
i 8.2"lm) to In to ^
Vm H-:-¿^
xi
1
"T^-
d|77. Vd
Use of Equation 2.71 yields
(2.131)
Sf < 8(4 + 2V2)Km)
m In m
TO +
■wt
For the sum Sg, one has
(2.132) Sg = Emin
1
1,
J=0
2xZ
m
(j
+
= 52 min / 1,
r)2 -
ß
3=0
2xí
—f —ß
and hence, the estimate obtained for Sf above applies also to Sg. It is now possible
to state the first main theorem.
Theorem 1. (a, to) = 1, to = 36, 1 = t < \/m,
=» Ep("/-«)p(^(i
y=o
\to
/
\to
+ r)2-^
0 = a < 1, 0 g /3 < 1,
y
< Vm[.8l(2 + V2)"(m>In2to + 33(4 + 2V2)Km) In to].
Proof. Lemmas 7, 8, 9, and 11 yield
R| < V
-^
(2.133)
[15-62 In2 Z + 8.10 In Z-ln to](2 + V2)
p(m)
+ 40(4 + 2V2)Km)[Vto— +, to In to
■wt
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228
DAVID L. JAGERMAN
Since x2 > 9.85, one also has
| R | < Vm[1.59 In2t + .823In Z-lnto](2 + V2)"(m)
(2.134)
+ 40(4 + 2V2)Km)
. to In to~|
V m -\
—— ■
id
Choose
(2.135)
Z = iy/m,
then, since In t < \ In to and x > 3.14,
| R | < VW-81(2 + V2~y(m>
In2to + 40(4 + 2V2)"<m)
(2.136)
+ 26(4 + 2V2)Km) In to].
One has
40(4 + 2V2)"(m)+ 26(4 + 2V2)'(m) In to
(2J37)
= (4 + 2V2)Km>In to • (**- + 26
\ln to
Since to è 36, In to ^ In 36 > 5.88, one has
(2.138 ) (40 + 26 In to)(4 + 2V2)Km> < 33(4 + 2V2)"M In to,
and hence
(2.139)
| R | < Vm[.81(2 + V2)'(m) In2m + 33(4 + 2V2)"tm>In to].
The conditions Z S: 3, to 2: 2x¿ are both met by the condition to ^ 36. Since to/2Z =
y/~m, the condition 1 ;£ r < to/2Z becomes 1 g r < Vro. The theorem is now
established.
The autocorrelation
IaÀ;
function of the sequence x¡ = i —j >is obtained immediately
from Theorem 1 by setting a = 0, ß = 0, and recalling that 4J(T) = R/m. Hence,
one has
Theorem 2. (a, to) = 1, to ^ 36, 1 Ú t < Vro
=* | iKt)| < m_è[.81(2+ V2)K"" In2to + 33(4 + 2V2)'(m) In to].
Consider the simultaneous Diophantine
(2.140)
0 g {- f < a,
(to J
inequalities
0 S<- (j + tY) <ß,
(to
J
Let the number of solutions of the inequalities
(2.141)
0 g j < to.
be designated by T(a, ß), then
G(a,ß) = TJ^Jl
TO
is the joint distribution
function of the sequences < —j2>A ~ (J + r)r-
sequences<-/>,<—(7
+ t)2> were independently
(to J (to
(to j (to
J
one would have G(a, ß) = aß. It is the present
equidistributed
object to determine
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J
If the
over (0, 1),
the devia-
AUTOCORRELATION
AND JOINT DISTRIBUTION
tion of G (a, ß) from the desired joint distribution
(2.142)
Ha(x)
229
OF SEQUENCES
aß. For this purpose, let
= a + p(x) - p(x - a),
then Ha(x) is a periodic function with Period 1 and, within the initial period,
Ha(x)
(2.143)
In view of the above properties
= 1,
0 g X < a,
= 0,
a á i < 1,
of Ha(x), the enumeration
r(«,ß)=Ep»(-/W-
(2.144)
j=0
\TO
/
a
T(a, ß) is given by
(J + tY .
\TO
Lemma 12.
r(a,f5) = *+íEp
m—1
3=o
\TO
/
Í-0
i=o
\m
-')
\m
P i" (J + r)2) -52 P(^ f - a) p (a (j + r)2)
/
\TO
/
y=o
52p^f)p(a(j+rY
i=o
.2
—J — a
TO
a E P ( -a (j + tY
+ r):
\m
+ 52 P ("A
j=o
\TO
a
-^Ep
m—1
/
+ «Ep(-0'
,=o
°/|
/
\TO
/
\TO
/
+ 52p(af-a)p(-(j+TY-ß).
\to
y=o
\to
/
Proof. Use of Equations 2.142 and 2.144.
Lemma 13. (a, m) = 1, to ^ 36, 1 á r < Vto, 0áa<
\TO
/
l,0á)3<l,
TU-1
I T(a, /3) - aßm I < 2 Zp(fl/)
3-0
E/
+
\TO
a .2
p (-j
,=0
/
-a
\TO
/
+
¿P
| y=o
-J
-
\TO
+ Vto[3.24(2 + V2)Km>In2to + 132(4 + 2V2)"(m>In to].
Proof. Theorem 1 enables the estimation of the sums of products of the p-functions to be effected. Also observing that
Ep(a/)
(2.145)
;=o
\TO
= E(^0+r)2),
/
y=o \to
/
and
(2.146)
52p(a:f
3)=Z(a(J
5=5' \to
/
+ rY -')■
3=0 \TO
the lemma follows.
In order to estimate
lemmas are required.
the sum of the p-functions
in Lemma
Lemma 14. (a, m) = 1, 0 ^ a < 1,
52 e ( — j2 — ha)
i=a
Vto
/
g y/2m(h, to).
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13, the following
230
DAVID L. JAGERMAN
Proof. Let
(ft, to) = d
h =d'
(2.147)
/
TO
" =d'
and
S = 52e(^f-ha),
(2.148)
3_0
\TO
/
then
fil = §«(*?/)
(2.149)
=d *¿V^is)
7=o \to J /
7=0
\to'
^ /
One has
m -1
m -1
/j,'
Is\2 = d252 52 er^f-^k2)
(2.150)
k=o j=o
\m'
7'
\
to
/
m—l
m —1+k
/i/
il
\
= d251
E efci2-^A
k=o j=k
\m
to
/
Introduce a new summation variable v by
(2.151)
j = k + v,
then
(2.152) | S |2 = d2 E1 E1 « (=*£ ,*) e (^
r=o fc=o \ to'
/
\to'
/) g d2 E1 ™E e (2-^ ,k)
/
£¡¡
feo
\ to'
/
By direct summation, one obtains
V
E e /2ft a
(2.153)
fc-0
7\
A
\ TO
= to ,
to \ 2ft a¡v,
to' | 2ft'aj\
Since (ft'a, to') = 1, to' | 2ft'aî< if and only if to' | 2v which may occur at most
twice. Hence,
(2.154)
| S |2 ^ 2dV = 2md.
The lemma follows on taking the square root.
Lemma 15. (a, to) = 1, to ^ 36, 0 ^ a < 1,
Ep (-/-<*)
\TO
j=0
/
<^
X
E Jft V(ft,to)+ 64.7Vto(4+ 2V2r(m)Inm.
láígíft
lgfiSi
Proof. Use of Lemma 1 yields
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AUTOCORRELATION AND JOINT DISTRIBUTION OF SEQUENCES
231
£»fe/-) <r \"t4-f)
3-_o
\TO
/
is*ätir/l
3=0
\TO
(2.155)
/
7«—X
1
l
+ E min / 1,
3=0
2xZ
a
.2
—J — a
m
Setting Z = § Vw aD-dusing Lemma 11, one obtains
E/fta
J - a
:&:>(=*
Vto
3-0
.5
el — J
""ft 3=o
lsiäävs
Vto
(2.156)
+8(4 + 2V2)v(m)Vto[~1+|lu
ml.
Since to ïï 36, one has In to > 5.88, hence
(2.157)
E -/
3=o Vto
A<e
/
» zc(iî/)
lSÄgeV-TTT»
3=0
Vm
/
+64.7Vto (4 + 2V2)Km)lnto
The lemma now follows on employing Lemma 14.
Lemma 16. (a, to) = 1, to 2: 36, 0 :£ a < 1,
m—1
E(a
3=0
/_
\
.2
|
< 64.9Vto(4 + 2V2)"(m)lnTO.
Vm
Proo/. One has
V2to
^
X
E £V(MÔ= ^EVd
l^figèV"»
ft
■""
d|7>.
E J
lakSivS"
(fl,m)=d
(2.158)
è^52Vl
T
d|m
52 I
lä*<V™
djÄ
ft
Let ft = cd, then,
^EVd
X
d|m
(2.159)
E
lgÄgiVmft
dlÄ
i^^TOE^
""
d|7» -V/d
E
laçai
iv^c
< Vm(.15 +.27 In to) E -i.
<*!"•
Vd
Use of Equation 2.71 now yields
(2.160) ^-^
IT
E
lSiSJVi
r V(ft,TO) < Vm(.15 +.271nTO)(2 + y/2)
ft
Since to ^ 36, one has
(2.161)
.15 + .27 In to < .296 In m.
and hence
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"(m)
232
DAVID L. JAGERMAN
(2.162)
y^m
yj
TT
1 ^/jj^
IgAásV»
< .296Vm(2 + V-V'"' him,
ft
Lemma 15 now yields
711— 1
/
\
I
52 p[-j2 - a) < .296Vn(2 + 2V2)Km)In m
(2.163)
3=0
\TO
/
+ 64.7Vto(4 + 2V2)r<ra) In to.
Since
(2.164)
.296(2 + V2)"(m)+ 64.7(4 + 2V2)"(">)< 64.9(4 + 2V2)"(m)
The lemma follows.
Theorem 3. (a, to) = 1, to ^ 36, 1 ^ r < Vto,
=> | C?(a,(8) - a/3 | < TO_è[3.24(2
+ V2)"(m) In2to + 392(4 + 2V2)'(m> In to].
Proof. Use of Lemmas 13 and 16.
Theorem 3 thus demonstrates
mately independently
that the sequences< - />,< —(j + r)2>are approxi-
equidistributed
(to j (to
J
Press,
Great
over (0, 1).
System Development
Corporation
Santa Monica, California, and
Fairleigh Dickinson University
Rutherford,
New Jersey
1. G. H. Hardy,
Divergent
Series, Oxford University
Amen House,
Britain.
1949.
2. Joel
N. Franklin,
v. 17, 1963, p. 28-59.
3. I. M. Vinogradov,
1954.
"Deterministic
simulation
of random
processes,"
Elements of Number Theory, Dover Publications,
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Math. Comp.,
Inc., New York,