On the Central Limit Theorem for the Prime Divisor Function
Author(s): Patrick Billingsley
Source: The American Mathematical Monthly, Vol. 76, No. 2 (Feb., 1969), pp. 132-139
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2317259
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19691 CENTRAL LIMIT THEOREM FOR PRIME DIVISOR FUNCTION 133
THE HARDY-RAMANUJAN THEOREM. If gm,, goes to infinity, no matter how
slowly, then
(2) D {m: v(M) - log log m }
(log log m) g2
If a set of density 1 is regarded as containing "practically all" integers, and
if we take gm to be, say, (log log m)1, so that gm(log log m)' is for large m very
small in comparison with log log m, then (2) does say that "practically all"
integers m have about log log m prime divisors. Thus an integer in the neighborhood of 108 will usually have about log log 108 - 3 prime divisors, and an integer
in the neighborhood of 1070 will usually have about log log 1070 5 prime divisors-remarkably few. (See [6] p. 358, where v(m) is regarded as a measure
of the "roundness" of m.)
In 1934, Turan [10] gave a greatly simplified proof of the Hardy-Ramnanujan
Theorem by an essentially probabilistic method. Further development of probabilistic ideas in number theory led Erdos and Kac [1 and 2] to conjecture and
to prove in 1939 a remarkable refinement of (2):
THE ERD6S-KAC THEOREM. If x_y, then
(3) D' x _("t) - log log M 1 ,
(3) Djm:x < (log
(lgoi)
e-u'du.
log nt)<Y
-\/2gr
This sharpening of (2) (we shall show later that (3) does imply (2)) shows
how v(m) fluctuates about the central value log log m. For example, for -x
= y =.9 the integral above is about .6, which is thus the approximate prop
of m for which the ratio in (3) lies in the interval [-.9, +.9]. If m is near 1070,
this ratio is approximately (v(m) -5)/51, which lies in [-.9, +.9] if and only
if v(m) lies in [5-.9X5i, 5+.9X51]- [3, 7]. Thus something like 60 percent
of the integers in the vicinity of 1070 have between 3 and 7 prime divisors.
(Since the abnormality of a finite stretch of integers can have no effect on the
density (3), there underlies a computation of this kind the premise that the
integers near 1070 are not atypical in their divisibility properties.)
Erdos and Kac in their original proof of (3) used difficult sieve methods.
The probabilistically most natural approach to the result is that of Halberstam
[4], which uses the method of moments (an idea first suggested by Kac [7]).
The purpose of this paper is to show hiow, by the introduction of further probability to avoid some heavy calculatioils, Halberstam's proof can be rnade more
transparent to the student of probability theory.
Preliminaries. On the space of positive integers, let Pn be the probability
measure that places mass 1/n at each of 1, 2, . , n, so that among the first
n positive integers the proportion that are contained in a given set A is just
P,(A) and hence (1) is the same thing as
(4) D(A) = lim P?(A).
n-*c
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134 CENTRAL LIMIT THEOREM FOR PRIME DIVISOR FUNCTION [February
Thus the Erdo3s-Kac Theorem states that
( V(m - log log m
(5) liM Pn M? () 3 < x} = (D)
n-- (log log m) I
where
(6)
(x)
1 er udu
=-
J
'dg
and the Hardy-Ramanujain Theorem can be similarly recast.
To see why (5) might. be true, observe first that
(7) v(m) - E MI m)
p
where 6p(m) is 1 or 0 according as the prime p divides m or not. For a positive
integer a, the number of multiples of a not exceeding n is [n/a], the integral
part of n/a, so that
(8) Pn{m: al m} Lil
which is nearly 1/a for large n. If pi, , pk
all i if and only if Ilipi m, so that the inter
has Pa-measure n-1 [n/Hipi], wlhich, for lar
of their individual Pn-measures. Thus, if m is chosen at random from 1 to n--according to P, that is-and if n is large, then the random variables
5,*,(m) are approximately independent, and there is som
that the sum (7) will obey the central limit theorem when properly normalized.
It is this sort of statistical plausibility argument that led originally to the
surmise that (5) might be true.
In the proof of the Erdos-Kac Tlheorem, we shall use from number theory
(in addition to the fundamental theorem of arithmetic) only the estimate
I
(9)
E-=loglogx+0(1);
see [6; p. 351], for example. From probability theory, we shall use the facts
embodied in the following four remarks, proofs of which can be found in Feller's
book [31 and elsewhere.
REMARK 1. If a random variable Dn converges in probability to 0, which will
be true in particular if E{ | } -*0, then a second random variable Un (on the
same probability space) has a given limiting distribution (say 4 as defined by
(6)) if and only if Un+Dn does [3; p. 247]. If Dn converges in probability to 0
and the distribution of UX converges to 1, then DnUlJ converges in probability
to 0 (since P{ Dn,Un| > j is at most P{ DThJ >e/x} +P{ I U,I >x}, i
limit superior is at most 2(1 -'b(x)); let x--*o). If An converges in probability
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19691 CENTNrA.,L LIMITT LrmOREMNI FORl PRIKMFE DIVISOR FU'NCTION 135
to 1 and Bn to 0, then Un has limiting distribution 4) if and only if A,Un+Bn
does (compare each with An U, = Un + (A n-1) U,) .
REMARK 2. Since 4) is determined by its moments
(10) gr =fJxrd f(x),
if distribution functions Fn satisfy ft x dFn(x)->fir for r=1, 2, . . , then
F.(x)->4) (x) for each x [3; p. 262], which is the basis of the method of momen
REMARK 3. If Fn(x)-4)(x) for each x, and if f .I xI r+EdF,(x) is bou
in n1 for some positive E, then f-,,0 xr dF (x)->ur [3; p. 245].
REMARK 4. If U1, U2, * * - are independent, uniformly bounded random
variables with mean 0 and finite variances O2, and if ff2 diverges, then the
distribution of Zt=l U/(= 2 converges to 4) [3; p. 258], a special case
of the central limit theorem.
From Remark 1 it follows that, if (5) holds for all x and g,-* oo, then
(v(m) -log log m)g;'(log log m)-1, regarded as a random variable under the
probability measure Pn, converges in probability to 0 as n-* oo, which implies
(2). Thus the Erdds-Kac Theorem contains the Hardy-Ramanujan Theorem.
Remark 1, together with the fact that log log m increases very slowly, can
also be used to cast (5) in a more convenient form, namely,
(11) lim p { v(M) - log logn < }
n o-~ co (log log n)
The equivalence of (5) and (11) will follow if we show that, for each positive E,
(12) lim Pn {r4 | log log mr-log log n >
n-+ co (log, log n)*
If n <m_n and the inequality in (12) holds, then
log log ni < log log n - e(log log n) *,
which implies log log n<e-2 log2 2. Therefore, for all n exceeding some no(6),
the probability in (12) is not greater than P, { m: m_nl }, which certainly goes
to 0. Thus the Erdos-Kac Theorem is equivalent to (11), and this is the form in
which we shall prove it.
Proof: first part. The heuristic ideas favoring the Erdos-Kac Theorem (see
(7)) figure in its proof as well. We shall compare the behavior of the 6,(m
that of independent random variables Xp (on some probability space, one
variable for each prime p) satisfying
1
1
P
P
(13) P{X, 11 = -I P{Xp. = 0l = 1 -
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136 CENTRAL LIMIT THEOREM FOR PRIME DIVISOR FUNCTION [February
The point of the heuristic argument is that the P.-measure of { m: B (M) -1,
i=1, . *, k} converges to P{Xi=1, i=1, * *, kl if the pi are distinc
Comparing the 3, with the X. indicates also where the norming constants in
(11) come from: If m<n, then no p actuially contributing to the sum (7) can
exceed n, so the distributions under P. of 1'(m) and of 5 b,(m) coincide; the
corresponding sum p5n Xr has mean i<n p-1 and variance "s< p-'
(1-p-'), each of order log log n by (9).
As the first step in the proof of (1 1), we shall show that it is unaffected if
we still further restrict the range of p in (7), replacing v(m) by
(14) vn(rn) = Z ap(m)
where { a } is a seqtuence so chosen that
(15)
,=
O(it)
for each positive e and
1
(16) - - o(log log I)'.
an<p? n P
The requirements (15) and (16) are met for examnple by
a,- =1/log log n.
this sequence goes to infinity slowly enough for (15), but because of (9),
quickly enough for (16).
For a function f of positive integers, let
I n
(17) Er{f} = - Ef(m)
n m=l
denote its expected value computed with respect to Pn. By (8),
En {Z } = p? P4m: 6p() = 1
p>an,
p>an
an<<p:Sn
P
and it follows by (16) and Remark 1 that (11) holds for all x if and only if
(18)
lim
Pn,{:
?x}
nr --+ oo(log log n)
=q(x)
does.
We shall compare (14) with the corresponding partial sum
(19)
Sn
=
E
X,p
P'a,
of the independent X, introduced above. By
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19691 CENTRAL LIMIT THEOREM FOR PRIME DIVISOR FUNCTION 137
(20) =ex, Z E -J Sn -1 2 1(1
P<?a
P
p<an"
p
p
which are the mean and variance of Sn, are each log log n+o(log log n)2, SO
that (18) is the same thing as
( (rnz) - c,, '
(2 t) lim Pn jIm: _ c - b(x)
n
-+oo
S,
Proof: second part. Since they depend only on Remark 1, the reductions
thus far (from (3) to (5) to (11) to (18) to (21)) present no technical difficulty,
although it is by no means trivial that such steps as the truncation in (14)which figures in the original proof of Erdos and Kac as well as in that of Halberstam-do advance the solution of the problem. It remains to prove (21), which
is the hard part.
Now (21) will follow by the method of moiments (Remark 2) if we prove
that, for r=1, 2,
(22) EnG(Vn -Crl,,r
converges to Actr (defined by (10)) as n-> o. To pr
we shall first show that its difference with
(23) El (Sn - C7)/sn}
converges to 0 (it is here we make rigorous the lheuristic argunment) and then
sliow that (23) itself converges to p,.
By the multinomial theorem and the definition (19), E{ S} is the sum
r ri ri rr
(24) E ' - - E" El{ XPi .. XPU}X
V-1 ri! .. r. *
where ' extends over those u-tuples (rl, , r,u) of positive integers satisfying ri+ - - +r,,=r and " extends over those u-tuples (pi, * *, p) of
primes satisfying Pi< < p <pu?<an. (We interpret a sumi as 0 if its range is
empty.) Since Xp assumes only the values 0 and 1, from the independence of
the Xp and the fact that the pi are distinct it follows that the summand in (24) is
(25) E{XP1 * Xpu
By the definition (14), En{z4J is just (24) with the summand replaced by
En{ ta'l * 5u }. Since bp(m) assumes only the values 0 and 1, from (8) and the
fact that the pi are distinct it follows that this summand is
(26) Enla= [1 - P. = E .
n -pi . .. p,
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138 CENTRAL LIMIT THEOREM FOR PRIME DIVISOR FUNC'i'ION [February
But (25) and (26) differ by at most 1/n, and hence EtS }
differ by more than the sum (24) with the summand replaced by 1/n. It now
follows by the multinomial theorem that
1 anV
(27) F{ S} - F" { v _- 1 - <n
$n
n
an inequality valid for r =0, 1, * Now
r
k
El(n-
Cn
k=O
and
En
{
k
(Vn
-
)rI
E
Cn)r
}
term for term and applying (27) we see that
k
I El (Sn - C) - En { (vn - CO) }I I < (r) n r-k = a+ CO)
k=O it n in
Since (a?n+Cn)r/n-->0, as follows by (15) and th
ence between (22) and (23) does go to 0.
There remains only the purely probabilistic problem of showing that (2
converges to ,Ur. Although this can be established by the sort of calculation
in the last century to prove the central limit theorem, here we shall deduce i
from the central limit theorem. By Remark 4, the distribution of (Sn -Cn)
converges to (. That (23) converges to A, will therefore follow from Remar
if we show that the moments (23) are bounded in n when r is increased to
next larger even integer. We shall in fact show that
(28) sup I El (Sn - Cn)7/sn} | < X
for every r.
Put Yp=Xp-1/p. By the multinomial theorem anld the indepen
the Yp,
(29) E { (Sn )r} = E r! - E{'YEl} * ... El Y. " I
U==1 ril r* Pi Pu
where ' and E" have the same ranges they have in (24). Since E
(29) still holds if we require in E' that each ri exceed 1. Since I Y
implies I E{ Yri }I I E { y2 , so that the inner sum in (29) has modulu
E,{ Yp} ... Et YP} < [ El EtVpI Sn
p M; yan
But if rl, - *, ru add to r and each is at least 2, then 2u <r. For n large enough
that Sn > 1, (29) now implies
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19691 CENTRAL LIMIT THEOREM FOR PRlME DIVISOR FUNCTION 139
r rlt
E(Sn - c.)J ?s, EZ'
from which (28) follows.
This completes the proof of the Erdds-Kac Theorem in the form (11). We
showed that replaclng v by P. as defined by (14) has no effect on (11), and we
proved the modlified (11) by the method of moments, showing that the moments
of v,n (normalized) are near those of the corresponding sum S.n defined by (19)
and that the latter moments converge to the p,.. We never did analyze the moments of v itself, although it is not hard to go on and do so (see [4]).
It is easy to show that the Erdds-Kac and Hardy-Ramanujan theorems
hold also if each prime divisor is counted according to its multiplicity: Let
a;(m) be the exponent of p in the prime factorization of m =ll p6v'(m) and define
V'(m) St O,(m). For k41, Q(m)-a(m) Ek if and only if pk+l jm, an event
which by (8) has P.-measure at most 1/pk+I; hence Er { a a- EtI t { m
8p(m)-bp(m)?k} ?2/p2, which impliesE.{v'-V} =0(1). It follows by Remark
I that (11) persists if r(m) is replaced by V(m), and by the same arguments as
before we can successively deduce (5), (3), and (2) with 9l(m) in place of (4
The arguments here suffice with little change, as in [4], for the central lim
theorem for a completely additive f for which f(p) is bounded and Ef2P
diverges. For an introduction to probability methods in number theory, see
81; for a comprehensive account, see [9
This work supported by NSF GP-6562 and Nonr. 2121(23), NR 342-043.
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