ON THE SIZE OF CERTAIN NUMBER- THEORETIC FUNCTIONS

ON THE SIZE OF CERTAIN NUMBER-THEORETIC FUNCTIONS

BY

WM. J. LeVEQUE(')

1. Summary of results. Let p denote a prime number, m and w positive

integers, and co, wi, w2 real numbers. Let/(?w) be an additive number-theoretic

function, so that f(mn) =f(m) +/(») if (m, ») = 1. Suppose that f(pn) =f(p)

and \f(p)\ =1. Then clearly/(m) = £p|mf(p). Let

An = ¿2í(p)/p, Bn = ( znpyp)1'2,pKn \ p<n /

D(u) = ■- f e-x2'2dx,(27r)»'2j_

and assume that 73„—► «> with ».

Erdös and Kac [7](2) have proved the following theorem: The number of

m^nfor which f(m) <An + o)Bn is nD(w)+o(n), as n—»°o. The present paper

is concerned with the proofs of a number of related results. In §2 there is

given a simpler proof of the special case of the above theorem in which f(m) is

taken to be v(m), the number of distinct prime divisors of m. The simplifica-

tion lies in that part of the proof using Brun's method; the central limit

theorem from the theory of probability is still used. Moreover, the error term

is improved, the term o(n) being replaced by O(»log3»/(log2»)1/4). (The sym-

bol log* » will be used throughout to denote the ¿th iterate of log «.)

It is shown in §3 that a similar reduction of the error term can be effected

in a theorem of Kac [ll], which says that the number of m^n for which

d(m) < 21°kb+"<1ok"»1'2

is »7>(w)+o(»). Here d(m) is the number of divisors of m.

It is probable that the error is actually 0(»/(log2 w)1/2), but this result

cannot be obtained without essential modification of the method used here.

§4 is devoted to a proof that/(?»),/(w + 1) are statistically independent,

with Gaussian distribution. This was stated without proof by Erdös [6].

In §5 this theorem is applied in proving that the number of m^« for

which v(m) <v(m + l)+o)(2 log2 «)1/2 is nD(w)+o(n). By the method of [ll]

it follows that the number of w = » for which

Presented to the Society, September 5, 1947; received by the editors May 13, 1948.

(') The author wishes to thank Professors M. Kac and J. B. Rosser for their invaluable

assistance in connection with the writing of this paper.

(2) Numbers in brackets refer to the bibliography at the end of this paper.

440

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SIZE OF CERTAIN NUMBER-THEORETIC FUNCTIONS 441

d(m) < 2"<2Io<ö">I/2d(w+ 1)

is also nD(o))+o(n). These results are generalizations of theorems proved by

Erdös [4]; these theorems are obtained by putting co = 0:

The density of integers m for which v(m) <v(m +1) is 1/2.

The density of integers m for which d(m) <d(m + l) is 1/2.

The following remarks might serve to orient the reader not already

familiar with theorems of the above type. Let/(m) be additive, and denote

by N(c, n) the number olm<n for which f(m) <c. The function xp(c) is called

the distribution function of f(m) if xp(— °o)=0, lK°°) = l, and if for every

finite c,

\p(c) = lim 7V(c, «)/».n—*w

Thus if f(m) has a distribution function, xp(c) is, for fixed c, the density

of integers for which/(ra) <c. Clearly not every additive function has a dis-

tribution function, since f(m) = log m has none. Erdös and Wintner [8]

showed that a necessary and sufficient condition for the existence of a

distribution function is that both the inequalities

f'(p) (f'(p))22_ -< «> and 2-,-< °°p p V P

hold, where f'(p) =f(p) if \f(p)\ ál and f'(p) = 1 otherwise. Hence if eitheror both of these inequalities fail to hold, we must adopt a different approach;

instead of speaking of the density of integers with a certain property we esti-

mate the number of integers which are less than » which satisfy a certain

condition, and this condition itself depends upon ». This leads, for example,

to the theorem of Erdös [5] that the number of tm = » for which v(m)

<log log w is w/2+o(»). The extension from this theorem to that of [7] cor-

responds, in the case of a function which has a distribution function, to an

extension from a theorem giving its value for some particular c to one ex-

hibiting the entire distribution function.

For use in §4 we now state a theorem of J. B. Rosser and W. J. Harring-

ton [H] which exhibits a result obtained by the use of Brun's method. It

may be put in the following form:

Theorem A. Hypotheses:

(a) A, Q are absolute constants.

(b) k, qx, q2, • ■ • , qr are positive integers relatively prime in pairs; ai

are integers with 0<a,<g,- for 0<i^T; ai} are integers, lî^T, l^j^ca,

such that for jy^k, aa^kaik (mod qî);f is a function having an integral value for

integral argument; Nt(k) = z2&(m), where the summation is over all integers m

such that simultaneously m satisfies some fixed condition independent of t, k, I,

q,-, an, at, and


442 W. J. LEVEQUE [July

f(m) = I (mod k),

f(m) p^ au (mod ft), 1 á * á ¿, 1 SS î S¡ <*<.

(c) ForO^fáT, Nt(k)^0.

(d) There are X, O O, independent of k, such that if we define Ft(k)

= (X/k)HUx d-ai/qt), then \ N0(k) -F0(k) \ <C for all k, I.(e) qx<q2< ■ ■ • <qr-

(f) IFe have chosen t, Y such that qt ̂ Y.

(g) There is an v, 0<v<l, such that for some x0, 103^x0ê(log r)V and

at log ?,1-,-Q log X

A log x

" (log2 *)2

/of aj>ô.

(h) For all i,0<at/qi ^v < 1, o a constant.

(i) 0.003eQlog2 F = 2.

(j) IFe have chosen w such that 1 <w, eQ log w^3(l — 77)/2.

(k) IFe are using the abbreviations

* a< 36/1(1 - t>) + 9.4Q + 9A2 36AZ=z2 — ;W = - -; 2 =

=1 ç; (1 - i>) log2 F Q log w log2 F

(1) Zg4<2(log2 F)/3.(m) IFe have chosen n an odd positive integer ^ 2.

Conclusion:r ¿W+iz+1 -^

iV((É) ZFt(k)U- (-!)«■(27rw)1'2e"(4 - w«(fc() log w)2)/4)

X 3Z(- 1)" T -JrTx-V ^ l0"2 F)/3

£ 4Q log2 F

(_1)nÇyn-l+2/(w-l)g. Z

With the same hypothesis except that in (m), "odd" is replaced by "even,"

the conclusion holds with "?g" replaced by "_".

In many applications of Brun's method the function g of (b) is defined by

g(m) = 1 ; in these cases Nt(k) is simply the numbers of integers in a certain

range having specified divisibility properties. The q's are frequently taken to

be the successive primes.

2. The order of v(n). The principal result of this section is contained in

the following theorem:

Theorem 1. The number of positive integers m^n for which

v(m) < log2 » + <o(logs »)1/2

is nD(œ)+0(n log3 »/(log2 w)1/4).


1949] SIZE OF CERTAIN NUMBER-THEORETIC FUNCTIONS 443

As was pointed out in §1, this can be regarded either as a special case of

the main theorem of [7] or as an extension of the principal result of [5].

The proof follows the lines of the latter paper; we shall preserve the notation

used there, making the following definitions:

1. T denotes the closed interval {log6 », »d0«* n)_3},

2. v'(m) the number of different prime factors of m which lie in T,

3- 2i, qi, • • • , qv symbols for the v primes q of T,

4. ax, a», • • • , the integers whose only prime divisors are q's,

5. a>x , a,2 , • • ■ , the integers whose factors are powers of k different q's

(k<2 login),

6. A(m) the greatest a,- dividing m,

7. Uk the number of integers m^n for which A(m) is an a[k\

8. ci, ci, • • • absolute constants,

9. *=D,l/3.It is known [13, p. 102] that Eps» l/> = log2 y+C+0(l/log y); since

x = ^p<n<io«2 »r31/p— X)p<iog6 n 1/p, it follows that

(1) x = log2 n — 4 log3 » + 0(1).

Lemma 1. 7*Áe number of integers m^n for which v(m)—v'(m)>(logi w)1'4

is 0(n log3 »/(log2 w)1/4).

We have (see [9, p. 355]),

¿ (v(m) - v'(m)) = E [~-l - £ Í-1m=X Pén Lp J q Lq J

= E T-1+ E [-1pSlog« n L P J n<>°K!»>-3<pSn L /> J

< » E -—i- mE —pglog» n /> fá» />

— n ¿2-1" »aogîn)~3p<n(log2 n)-3 ^>

= »{logs n + log2 » + 3 log3 » — log2 « + 0(1)}

= 0(n log3 »),

and similarly,

n 111

E M#0 — »»'(>»)) > » E —h«E — n E_ — i°s6 w — nm-l j>álog«n Í pén P pS»««») />

= 0(» l0g3 «).

Hence the number of integers m^n such that



v(m) — v'(m) > (log2 m)1'4

is

0(n log3 n)

(log» n)1'*

which gives the lemma.

Lemma 2. IFe have

xk 3 _, 1 ¡s*-< >2 — <—>kl log6 n i af k\

where the dash on the summation means it is extended over the square-free af

only.

The proof is as in [5].

Lemma 3. Uk = ne~xxk/k\+0(n/log5 n).

The proof is as in [5 ].

Lemma 4. The number of integers m^nfor which v'(m) <log2 » + w(log2 w)1/2

is nD(oi)+0(n log3 »/log2 »).

Clearly v'(m)=v(A(m)). Let us first consider the number of integers

«Í» for which v'(m) <x+ux112; this is

E Uk.k< z+a è11

We put y=x+wx112. By Lemma 3,

k<y k<y kl \l0g6 »/

It is known [l ; 10] that there is a constant Ci such that

Cl

r- E ^t - -D(«)<gllS

and consequently

•^ / ny \E Uk = «D(«) + 0(»/x"2) + 0 —- )

(2) k<y Vlog8 »/

= m7)(co) + 0(n/x1'2)

since #~log2 ».

We now consider the integers m^n for which

x + ux112 < v'(m) ^ logü » + œ(log2 »)1/2.



Since xk/k\ assumes its maximum value for k= [x], we see by Lemma 3 that

the number of integers m^n for which v'(m) =k is at most

(3) ne-*— + o(——)<\ log » /

xx / n \ c2n—+ 0(-) <-x\ \log»/ X112

Hence the number of m ^ » for which x+wxl,i <v'(m) = log2 « + w(log2 »)1/2

is at most

c2»— ((log, n- x) + co((log2 nY'2 - x1'2)).X1'2

By (1), this is

0(^-(l0g3 n + „-^-■)) = o(-=^-Y\*1/2\ (log, »)1/2 + *1/2// \(log, »)1/2/

This together with (2) completes the proof of the lemma.

We come now to the proof of the main theorem. By Lemma 4, we have

only to prove that the number of m^n for which

v'(m) < log2 n + o)(log2 »)1/2

and

v(m) ^ log2 « + w(log2 n)l,i

is 0(n log3 «)/(log2 »)1/4).

We divide these integers into two classes; in the first class are those for

which

v'(m) < log2 n + w(log2 »)1/2 — (log2 »)1/4,

of which there are 0(» log3 »/(log2 »)1/4) by Lemma 1, and those for which

log, » + <d(log2 »)1/2 — (log, m)1'4 =■ v'(m) < log2 » + co(log2 »)l/2,

and on account of (3) there are only

o((\og2 n)1«-—} = o(-^\V x"2J \(log2«)i/4/

of these. This completes the proof of the theorem.

3. Application to d(m). We now prove the following theorem.

Theorem 2. The number of integers m^nfor which

d(m) < 2logî n+"(lo«2 n'1/2

is also nD(w)+0(n log3 «/(log2 w)1/4).

Let k„(w) be the number of m^n for which v(m) <log, » + w(log2 «)1/2,



rn(u) the number of tw^» for which d(i») <21<J« n+"(l0^n)1/2, and p(n)

the number of »î = » for which ¿(w)/2"(m) <2t(log2 ")12. Then we have, as in [3],

kn(u — t) — (n — p(n)) ^ rn(co) ^ kn(o>),

for every e>0. On account of Theorem 1, we have only to show that

»log3 »kn(o) — e) = kn(w) +

and that

In log3 m \

\ (log, »)»'«/

/ » log3 M \- i(») =0(- )

V (log, «)W

for suitably chosen e.

We define

(1 if1 if k\m

k\ m.

Then clearly

Since

we have

d(m)

d(m) = IT (1 + P,(») + PÄ™) + •••),p

tHmi = II (1 + P,(»)).

1= 1 - —pp(m),

1 + Pp(m) 2

~" n {(i - ~ p,(»)) (i + p,(») + p^(») + •••)}

= YL] ! + Pp(m) + pAm) + • • • - pp(m) - — Pp*(m) -

= Il{l + y PP2M + jPAm) + •••}•

Multiplying out, we get

- = 1 + ¿J -2»(>») *r 2"t/!>

where the asterisk indicates that the summation is extended over all k which



are such that if p\ k then p2\ k, for every p. Hence

¿ *Ü_n+2Z.J-\!L\<n + „x*_L-,±* 2»(«) ^ 2"<*>L¿J *-2'<*>

and we represent this convergent series by 2*, so that

" d(m)E —L-L<n(l +2*).±1 2'<«>

Now if there are ¿u integers m such that 0<m^n and

2Kw>)

then

> 2«(loS2 K)l/2

hence

E ^-V>«-M + i"-2íaogíB) ;

« - M + íf2«<l0« »>lft < «(1 + 2*),

from which we get

»2*

** 2,(lot2 n)1/2— 1

Consequently

and so

/ 2* \») = » — ¡i > ni 1-ru-),

' \ 2«<log2n) — 1/

«2*n - P(n) < „.„.m-

2«(iog2 »r* — i

We now take

logs ne = -

log 2 log2 »

and get

n — p(n) <* / n log3 » \

~1 " \ (login)1'*)'log2 » — 1 \ (lcg2 n)1

Finally, it is clear that



/ » logs » \*,(« - i) = *,(«) + 0(e») = k„(o>) +0 ).

\ (log, «)1/4/

Thus we have proved Theorem 2.

4. A general theorem. We shall now prove the following theorem.

Theorem 3. 7e/ f(m) be a strongly additive function, that is, f(mn) =f(m)

+/(») if (m, ») = 1 and f(pn) =f(p). Suppose that \f(p)\ =1 for all primes p.Let 2^2pP(P)/P= °°i Qnd let cox, w2 be any real numbers. Then the number of

positive integers m^n for which simultaneously

f(m) < An + o)xBn and f(m + 1) < An + a2B„,

where An= z2*û»f(P)/P, Bn=(2~2v^f(P)/P)m, « n-D(ut)D(ut)+o(n).

The argument used in the proof is a direct extension of that used in [7].

Lemma 1. Let fi(m)= ¿2p\m,p<i f(P)- Then denoting by o¡ the density ofpositive integers for which simultaneously

fi(m) <Ai + wxBi, ft(m+l) KAt + uiBt,

one has limi->« 5¡ = 7)(coi)7>(ío2).

Let

Then clearly

Pp((f(p) if p\n,

n) = \I 0 if p\n.

fi(m) = ¿2 Pv(m)-p<i

We divide the proof into two parts.

I. The functions app(m)+bpp(m + l), where a, b are any fixed constants,

not both zero, are statistically independent. To prove this, it suffices to show

that

3/Í gi2pep(app{m)+hpp(m+X)) X — 7T ]ß j eHapp<.m)+bpp(m+X)) )

pGp

where M{t(m)} =lim„^00 E«-i t(m)/n, and P is any set of primes all less

than /. We give the proof only for the case where P consists of two primes p

and q; the argument is no different in any essential respect in the general

case. We prove then that

fyf f gi(app(m)+bpp(m+X)+apq(_m)+bpq(km+X)) I

(4) '= MÍ ei(a'l!>(m>+4',!>(m+1)) ] . M l gi(apq(m)+bp,(m+X))\

Clearly,



giaa __ 1

giapp(m) = 1-)-Pp(m),

a

where a=f(p), and similarly for e^V"0. Hence the left side of (4) is

(/ eiaa — 1 \ / eiba — 1 \

mUí +- P,(»)Ji 1 +-Pp(m+ 1)\

I gieß _ ! v , gibß _ 1 VN

•il +-P«(»)jí 1 +-P,(» + 1) H •

Let us put

giaa _ 1 gib« _ 1 gia/3 _ 1 g»&0 _ j

a a ß ß

then the product in the braces is, since pp(m)pp(m + l) =0,

(1 + ApPp(m))(l + BpPp(m + 1))(1 + ^,Ps(«))(l + BqPq(m + 1))

= (1 + Appp(m) + BpPp(m + 1))(1 + Aqpt(m) + BqPq(m + 1))

= 1 + ApPp(m) + BpPp(m + 1) + AqPq(m) + BqPq(m + 1)

+ ApAtf^m)^™) + BpBqPp(m + l)Pq(m + 1)

+ ApBqPp(m)Pq(m + 1) + AqBpPq(m)Pp(m + 1),

and the mean of this expression is

1 + aAp/p + aBp/p + ßAq/q + ßBJq + (aß/pq)(A^4.q + BpBq)

1 " 1 "+ ApBqlim — X Pp(ni)Pq(m + 1) + AqBp lim — 2~2 Pp(m + l)p9(w).

n-»« » ,n.= l n-»w » m=i

We have

aß if m = 0 (mod p) and m = — 1 (mod ç),Pp(m)Pq(m +1) = \

\ 0 otherwise.

The general solution of the pair of congruences m = 0 (mod p), m = — 1 (mod q)

is m = m0+tpq, where mo is the smallest positive solution, and there are

1+ [(» — mo)/pq] solutions less than ». Hence

(F n — m0~]\1 + 1-J W,

and since 0<m0<pq,

aßrnl 1 " aß aßT »"1- - <-Ep,Wí,(* + 1)<- + - - ,» LpqJ « m«l » » L^çJ



so that

1 " aßAPBgApBq lim — 2^ Pp(m)Pq(m + 1) =-— •

b->» n m=i pq

Similarly,

1 A aßAqBpAqBp hm — 2-, Pp(m + OPeW =-'

»->» » m-i pq

Therefore, the mean of the left side of (4) is

(1 + aAp/p + aBp/p)(l + ßAJq + ßBq/q),

and this is obviously the same as the mean of the right side.

II. We can now prove that

MÍgit(fl<.™)-Al)IBl+iv(fl(m+X)-Al)IBl\ _> ¿-(.f+^H

as I—* oo. For clearly

M j eH(/l(.m)-Al)lBi+iv(fl(m+X)-Ai)IBi\

(5)= g(-<A¡/Bí)({+>l)Jlf { e<*/*l>«/íC«>-H/l(m+l)}

g¡({/l(m)/B¡+7!/¡(m+l)/B¡) = eH(2r£,ppím) /Bi+^g^tm+D /S¡)

= H ei({"pc",)/-Bl+"'i,(m+1)/-Bl).

PEÍ

Hence

Jlf íci(í/l(m)/Sl+ii/l(m+l)/í<) = jlf J ]T ei(.(pp(.m)IBl+,pp(m+X)IBl){. ̂

\ pal )

and since by I the numbers i¡pP(m)/'Bi+ripP(m + l)/'Bt (p^l) are independent,

this is

J M•|ei(f''ptm)/JS,+'",p(m+1)/s,) >

/ eiU(p)IBi _ 1 g¡i/(p)/B¡ _ 1v

= n(i+—-—+■—-—)Péi\ p p /

( ^ ( if(P) 1 P(P)= exp \ E log ( 1 + -^ ({ + i») - - 7-fJ (£* + ri2)

7> —-)}



By taking / sufficiently large, we can make

if(P) 1 f(P) i P(P)^(^ + V)-T ^ (e + r,2) - - ^ (? + „») +pBi 2 pB, 6 pB¡

<1,

since 73j—>œ with 7 Then

iJ(P)— (f + ij) - -pBt W T W 2 pB\

( if(P) 1 P(P) \

if(p) 1 f2(p)

2 \pBi

1 /*(#)

/ pBi+

so that

Eiog(peí \

if(P)-(í + uH ••• J

Í(Í + U) y /(#)_ _ J_ if + I?2) £ A#)

73j péi p

i^ V"iT(? + ,?)_

Bi pî p

1 f(PW + v2)

2 Bi+

-7Ï2 p=¡ />*

E —

y+

Bi

ps¡\ ^ /

+

i v (¿/wtt + n) — y2B\hi P> '

and all terms of this expression approach zero except the first two. Conse-

quently, by (5),

lim M{eX-ilBl)(.(lUm)-Al)!;+v(jam+X)-Al)) J = g-(£2+,2)/2_

Lemma 1 now follows immediately from the continuity theorem for

Fourier-Stieltjes transforms (see [3, pp. 96-102]).

Now let $(») be a positive function which tends to zero as »—><» in such

a way that

l/$(») = o((log log n)2)



and also

l/$(«) = 0(Bn).

1/2Let »*(B)=a„, »»(»» =ßn; clearly an—>°o, /3„—►«> as «—>°o. Let ai(«),

a2(»), • • • , be the integers whose prime factors are all less than an, and let

\p(m, n) be the greatest a,- which divides m. Let <7i, g2, • • • , qr. be the primes

less than an. We have the following lemma.

Lemma 2. 7"Ae number of positive integers m^n for which

ip(m, n) = aK(n), xp(m + 1, n) = ax(n),

where aK, a\^ß„, is zero if (aK, a\) >1 or if 2¡¡aKa\ and otherwise is

—-—rî(i—) n fu-Vi + o(i)),44>(aKa\) ¿_2\ qj i&i&Tn.q¡\aKo>¡ \ ?.(?< — 2)/

where <p is Ruler's <p-function.

This is clear if (aK, a\) > 1 or 2!¡ata\. Hence assume that (a„ a\) = l, 21 aKa\.

Then there is a unique integer r0, 0 <r0<a, such that

(6) r0-aK = — 1 (mod a\).

Let

r0a, + 1(7) -= b,

a\

and consider the numbers r of the form

(8) r = r„ + ga (g = 0, 1, 2, • • • ).

ra« + 1(9) -= b + gaK.

a\

We wish to count the integers m^n for which

m = RaK, m + 1 = Sa\,

where neither 7? nor S has any prime factors less than an. If this is the case,

it must be that

Ra, + 1=0 (mod ax),

and 7? must be of the form of the r of (8). So we consider the numbers r and

(raK + l)/a\, and ask that

raK + 1r fé 0 (mod q%), -^ 0 (mod ?,) (* = 1, 2, • • • , Tn).

a\

From (8) we require then that



(10) ga\ fé — r0 (mod g.) (i = 1, 2, • • • , Tn),

and from (9)

(11) gaKfé — b (mod g<) (i = 1, 2, • • • , 7"»).

If qi\ Ox, (10) holds for every g, by (6) ; and if g,| a«, (11) holds for every g,

by (7). If qi\a\, (10) is equivalent to some restriction

g fé e, (mod g¿),

and if q<\at, (11) is equivalent to some restriction

g fí ft (mod fi).

Moreover, e,-^/,- (mod g,-), for if the opposite were the case, we would have a

g' such that

g'ax = — r0 (mod g,), g'aK = — 6 (mod g,),

and by (7),

r0a, + 1g'aK m-,

a\

g'aKax = — r0ö« + 1,

— f0a, = — r0aK + 1,

1 =0.

So we must count the integers g such that

» ro(1) 0 < g g- (since m = (r0 + ga\)at g »),

a«öx ox

(2) g - 0 (mod 1),

S 7e «h fi (mod g,) if 1 á ¿ = 7",,, g< | a«ax,

(3) g fé a (mod g¿) if 1 á i á Tn, g¿ | a«, g4ax,

g fé fi (mod g¡) if 1 ^ i ^ T„, q{\aK, g¿ | ax-

We define Nt(k), 0^t^Tn, to be the number of integers g which satisfy

(1) and (3) (with T1,, replaced by /) and

(2') g - / (mod k),

where 0<Kk, (k, gO = l for i = l, 2, ■ ■ ■ , t. Thus there are 7Vrn(l) integers

of the kind specified in Lemma 2.

We also take, for 01117,,

» ' / a,\Ft(k) = -—n(i--),

kaKa\ ¿_i\ qj



where

¡1 if g<| aKax,-Í(2 otherwise;

in particular, ai = l (an empty product is unity, as usual). Then

t» — ajo 1—u-kaKa\ J

and

wFo(k) =

kaKa\

so that

| N0(k) - FQ(k) | < 2.

We now apply Theorem A; the following remarks and definitions apply to

the corresponding hypotheses:

(a) We take A = 1, (9 = 2.

(b) We take T=«¡, f(m)=m, g(m) = l. (We have replaced m by q.)

The fixed condition is

\ aKa\ /

(d) C = 2. '

(f) t = w(an) = Tn, Y = qTn.

(g)

^ a, log g< log qiS=J2 -— - Q log x < 2 2Z -^~ - 2 log x

«¿s x qi g,g x qi

/log x 1 3 \< 2(-^ + — + —)<5.

\ j x 2/

Since the number of a's which are one is less than log X,

c^ oV log g¿ ^ log g» ,,S = 22^-¿^ -• - 2 log a;

«ai g¿ ¿-i g¿

log x log2 X 1 3> - 2 —5-3 - log, X

log X log X 2

> - (2 + 3 + log2 X+1+ — + — ) = - (8 + log2 X),

so that



| SI < 8 + log2 X.

Taker) = 2/3, Xo = e<-l°**x>\ Then xB<e<~l°* «'. Assume *(»)>(log, X)3/log X.

Then if x>xe, X>Xx (that is, w>»i),

/ log, x y8 + log2 X < (-) =

\2 logs X

log xo log x

logs X) (log xo)2 (log2 x)2

and hence

i log x\S\ <---

(log, x)2

for x>xo.

(h) Take v = 2/3.

(i) The condition 0.003 eQ Iog2 F>2 is certainly true for X>X2 (that is,

«>»,).

(j) Take Kw<6/5.(k) We have

5 117 18Z ^ —• log2 7r(x*), IF ~-> 2 ~- •

2 log2 X log w log2 X*

(m) We shall replace nby v; then p = l (mod 2), v>2.

Since all hypotheses are satisfied, we infer the conclusion of Theorem A,

for »>max (»i, w2). In order to obtain an asymptotic expression for Nt(k),

we must show that, for suitable v, w, as «—-><»,

gir+22+2

>0;

this implies that

(2TTVyi2e'(l - w«e2Ç}2 log2 w/4)

g(log w)—1

^0.„l/2gv

II. —(-) = o(Ft(k));k \4Q log2 Y13)

this implies that

X-= o(Ft(k)).(S-logX)8'3

III. CF«-1+2/("'-1)ez = o(Ft(k));

this implies that

qn-x+2nw-x) log5/2 „.(X*) = o(F,(k)).



To satisfy these conditions, we choose

f[l/$1/2] if this is odd,

ll + [l/f»1'2] otherwise

and

W = 1 + $1/2.

I. We have

edog«)-1 e(ioE(i+*1/i))-1 g»1'2*'2)-1

--= --r¡r- <-=T75-*1/2 = 4.i/2ei/,+*1/2/4+..._^0-vl/2gv $-l/2g* "2 g* m

II. Clearly

Ï '/ 2\

« »-A g.7

and it is known that for some constant c>0,

Ú(.-i)í<-i \ g¡/?¿/ (log g*)2

Hence

cXFt(k) =

¿3>2 log2 X

and therefore

X- = o(Ft(k)).(4> log x)8'3

III. gn-l+2/(K,-l) l0g5/2 n-(Ar*) g X*(*_1/2-1+2*_l,2) log 6'2^4>

g X3*1'2-* log5'2 X = o(X{)

for every 5 >0. Hence this term is also o(Ft(k)).

Collecting these results, we infer from Theorem A that

N,(l) <Fl(l) + o(Ft(l)),

and by making an obvious change in the definition of v so as to make it even,

we deduce that

Nt(l) > F,(i) - o(Ft(l)),

so that finally

NrnW > Fr.(l) - o(FTn(l)).



But

«D-jrltfi-•=!)--=- n(i~=0i-i\ gt/ a«ix í=i\ g</

n (>--)n jV/. 2\ 2áiá»,o4-=i V g</

2a,ax ¿-2 V g,7 -^ / 2 \

2â»ëi,ai=i\ g¿/

= ~n(i-^) il (1+-4)¿aKa\ i=2\ g,/ iéiút,qi\aKax\ gi — ¿/

n (i-i)2a¿áí,5j|a,ax \ Ci/

n (i-i)

i<j>(aKa\) i=2\ g,/ sSiS*,«!«.»«^ g<(g¡ — 2)/

and putting / = Tn we have the lemma.

Lemma 3. 77îe number y of integers ^M divisible by an a,(«) >ßn is less

than è il7( <£(»))1/2, where b is an absolute constant. (It follows from this that the

density of integers which are divisible by an di(w) >/?„ i* less than b($(n))1/2.)

This is Lemma 4 of [7].

Lemma 4. Denote by ln the number of positive integers m^n for which simul-

taneously

(12) /„„(«) < /L„ + coi73a„, fan(m +1) <Aan + a,,73a„.

7/se» /„ = nD(ux)D(o)2) +o(n).

Divide the integers ?w=w which satisfy (12) into classes Exx, Exi, Eix,

E13, En, E3x, • • • so that mCEEa if and only if simultaneously

\p(m, n) = a,(»), ip(m + 1, ») = a¡(n),

and denote by \A\ the number of elements of A. Clearly

in = 2Z\Eij\= E I&/1+ E |ä/|.i,¿ <riiO|,a/£Ai >'.j';<i,->jSn or ay>3„

By Lemma 3,

E |£«| < J«(4>(«))"2,Ci>ßn or aj>/S„



and therefore it suffices to show that

J2 | Eii\ = nD(o>x)D(u2) + o(n)

as »—» oo.

By Lemma 2,

E |a,l-|ft(i--) E'^p^, . ai.ajäßn 4 4_2 \ g,-/ n,,a,â|S„ ^K^í)

+ .(.ñ(i-2-) £'^^).\ »-2\ g.7 a„aj£ßn <t>(aiOj) /

where the dash indicates a summation over the a,-, a,- satisfying

/«„(«<) < ^4a„ + wi5«„, /<»„(«,) < Aan + co2Ban, (a„ a¡) = 1; 2 ¡ a«!,-,

7J(«i, a/, ») = II ( 1 + —,-rr) •si«ST..S4|a<«,\ Çi(g.- - 2)1

Now divide all the positive integers into classes Fxx, F12, ■ ■ ■ such that

mÇzFij if and only if xp(m, »)=a,(»), xp(m + l, n)=as(n), and let {Fij} be

the density of T",-/. Now consider the set E'T7»;, with the dash as before.

Putting l = an and using Lemma 1 we have that

{E'T7.;} = D(»t)DM + o(l)

as «—► oo.

Now

(14) E'F.-i = E' *« + E' *«.a,-,a,S0n a,->0„ or a/>0„

and by Lemma 3,

(IS) { E' F«l < 6($(»))i'2.Va¡>(3„ or aj>ß„ )

Furthermore, there are only a finite number of ö's which are less than ßn, and

hence

{ E' f<\ = E' {Fa}.voi,a,SS„ / ai,a,ußn

But

' 0 if (au a¡) > 1 or 2\ a,<z;,



and hence

l _ 1 n Si- / 2 \ _ P(a,-, a,-, »)(i6) { E' ¿a =-n(i--) jy \ \ •

Combining (14), (15), and (16) we have

-mi-- E' \; ; = DMDM + od)4 ,_2\ g,7 at.ajSßn 0(«««i)

as w—»oo. This and (13) give the lemma.

The remainder of the proof of Theorem 3 is almost identical with §5 of

[7].From Theorem 3 we get the following corollary.

Corollary. The number of positive integers m^n for which

v(m) < log log » + wi(log log »)1/2,

v(m + 1) < log log » + co2(log log «)1/2

is nD(ci)x)D(wi)+o(n).

5. Applications. In this section we apply the corollary stated at the end of

§4 to theorems concerning the relative sizes of v(m), v(m + l) and of d(m),

d(m + l).

Theorem 4. Let tn(u>) be the number of positive integers m^n for which

v(m) <v(m + l)+o}(2 log log »)1/2. Then tn(os) =nD(w)+o(n).

Let us put Xn(m)=v(m)—\og log »/(log log «)1/2, l=w¿», then the

corollary to Theorem 3 can be put in the form

lim„^M Prob {X„(7»)<wi;X„(w + l)<w2}=7»(wi)7»(w2).

Taking the characteristic functions, we get

1 " / / v(m) — log2 « v(m + 1) — log2 »\\lim —■ y. exp I i ( £-h tj-— ) 1_» »rr, \\ (iog2«)1'2 dog,»)1'2 //

1 /■"/" / x2 + y2\

= 2^J_J_.eXPx-F" )dXAy'

Taking 77 = — £, we have

,. 1 " / v(m) - v(m + 1)\lun — 2^ exp ( li.-—-1n^» » «_! \ (log, »)1/2 /

(17)1 c °° rM / x2 + y2\

= — I I exp (t£(* - y)) exp Í-1 dxdy.



Weputí¿(w)=ín(w/21'2);if

tn'W)

1 " / v(m) - v(m + 1)\ Pxlim — E exp ( ¿f-—— ) = e*'p(x)dx.»-» » m_l \ (10g2 n)11' / J-oo

l{U) C i XA-► I p(x)dx,n J -x

then

v(m) — v(m + 1)^

(18) n-.» » ¿TÍ V ' (log2 M)1

Comparing (17) and (18), we see that we must write

1 r °° ç M / x2 + y2\— I J exp (¿£(x - y)) expi-jdxdy

in the form

/e^zp(x)dx.-00

Obviously

1 /•M /• °° / »2 + y2\— J I exp (if (a; - y)) exp (-J ¿xdy

If00/ .-r2\ If"/ A= ̂ r^J-exp v* ~ TJîi^J-exp r * ~ TJrf>'

= (- f expf-(*2 - 2t{* - f2)-)dx)\(2*yi2J-x F\ 2 2/ /

/exp(-f2/2) /•- y= I- I exp (— u2/2)du 1

\ (2t)1'2 J_„ /

= exp (- f2).

Hence we must find a p(#) such that

e-f = I e^xp(x)dx.J _oo

Since

- f e«v-*'/*¿* = e-f2'2,

we put $ = 21/277, x=y/2112, and get



,>-¡/2/4

— f(2try'2J-x 21'2

Hence p(x) = I^t^'V"*2'4, and

dy = e= e-i

which implies that 2„(co) =»7>(w)+o(«). This completes the proof.

Theorem 5. 7e¿ rn(oi) denote the number of positive integers m^nfor which

d(m) < 2"<210*2 ^ll2d(m + 1).

Then rn(u>) =nD(w)+o(n).

We make the following definitions:

d(m)2"<-">+»f(m) =-1

d(m+ 1)2'<»>

g(n) = 2«<21°s2«>1/2 (t > 0),

Fn = E ¡0 < m =: »; v(m) < v(m + 1) + (u - e)(2 log2 »)1/2j,m

Gn = E {0 < m = n;f(m) ^ g(»)},m

En = 7i {0 < m = »; d(w) < 2"<2 >°s2 »)1,2<f(w +1)},m

#(«) = | G„ |.

If 7» is in both Fn and Gn,

d(m + l)2"(m>g(w) ,,.d(m) < —-— < d(m + 1)2»<2 ** »i1'*,

so that mEiHn, and therefore 7„G„C77„. Since |7n|=i„(w —e) and |77„|

= rn(w), it follows that

(19) U(u - e) - (n - p(n)) = rn(œ).

Since d(m)/2v(m) = 1, we have

1 " 1 " d(w)— E/(™)= —E —>n£t n ±i 2'<«>

so that (see [ll])

1 "lim sup —• 2~2 f(m) < °° •

»->» » m=1

Consequently p(n)/n—>l as »—>oo, and from (19) we get



(20) tn(o - e) - o(n) = rn(co).

Now let F(m) be the number of (not necessarily distinct) prime divisors

of m ; it is easy to show that 2'(m) ̂ d(m) ^ 2-P(m). It is known (see [9, Theorem

435]) that there are constants B and Csuch that

n n

2~2 v(m) — n log2 n + Bn + o(n), E F(m) = » log2 n + Cn + o(n).m=mX m— 1

Hence if 6(m) =F(m) —v(m),

— ¿ 6(m) =C- B + o(l),n m=i

so that 6(m) has finite mean.

We now put

in(w) = I E {0 < m ^ n; v(m) < F(m + 1) + oi(2 log2 n)U2} \m

£JÛ<»a»; v(m) < v(m + 1)

/ 6(m + 1) \ )\+ ( w + —-)(2 log2 n)1'2}].

\ (2 1og2»)W j

Clearly

(21) r.(») á «.(«).

Let

An = E [0 < m ^ n; 6(m + 1) > log3 »} ;

since 6(m) has finite mean, \An\ =o(n). Moreover,

( / 6(m + 1) \ 1E {0 <m¿ n;v(m) <v(m+ 1) + ( a> + ——-•) (2 log2 m)1'2^m ( \ (21og2«)1/2/ )

= E \0 < m S n; v(m) < v(m + 1)

/ 8(m + 1) \ ■ 1

+ E <0 < m g »; j»(>») < »»(w + 1)

/ e(m + 1) \ )



The number of elements in the first of these sets is o(n), and the number of

elements in the second set is ¿„(oj+e'), where

<> = log3W_»0.

(2 log2 w)1'2

Hence

s„(u) = t„(w + e') + o(n),

and this with (20), (21) gives

tn(oi — e) — o(n) g r„(co) ^ tn(o> + e') + o(n).

Since í is arbitrary and e'—»0 as w—>», we conclude that rn(u>) =nD(co)+o(n).

Bibliography

1. A. C. Berry, The accuracy of the Gaussian approximation to the sum of independent

variâtes, Trans. Amer. Math. Soc. vol. 49 (1941) pp. 122-136.

2. V. Brun, Le crible d'Eratosthene et la théorème de Goldbach, Slcrifter Videns, Kristiana,

1920.3. H. Cramer, Mathematical methods of statistics, Princeton University Press, 1946.

4. P. Erdös, On a problem of Chowla and some related problems, Proc. Cambridge Philos.

Soc. vol. 32 (1936) pp. 530-540.5. -, Note on the number of prime divisors of integers, J. London Math. Soc. vol. 12

(1937) pp. 308-314.6. -, On the distribution function of additive functions, Ann. of Math. vol. 47 (1946)

pp. 1-20.7. P. Erdös and M. Kac, The Gaussian law of errors in the theory of additive number-theoretic

functions, Amer. J. Math. vol. 62 (1940) pp. 738-742.8. P. Erdös and A. Wintner, Additive arithmetical functions and statistical independence,

Amer. J. Math. vol. 61 (1939) pp. 713-721.9. G. H. Hardy and E. M. Wright, The theory of numbers, Oxford Press, 1938.

10. P. L. Hsu, The approximate distributions of the mean and variance of a sample of inde-

pendent variables, Ann. Math. Statist, vol. 16 (1945) pp. 1-29.

11. M. Kac, Note en the distribution of values of the arithmetic function d{m), Bull. Amer.

Math. Soc. vol. 47 (1941) pp. 815-817.12. -, Note on partial sums cf the exponential series, Revista de la Universidad

Nacional de Tucumán, Serie A, Matemáticas y Física Teórica, vol. 3 (1942) pp. 151-153.

13. E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, 1909.

14. J. B. Rosser and W. J. Harrington, Brun's method in number theory, to be published

soon.

Harvard University,

Cambridge, Mass.


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