ONE-MACHINE SEQUENCING TO MINIMIZE CERTAIN FUNCTIONS OF JOB TARDINESS by Hamilton Emmons
ON THE SIZE OF CERTAIN NUMBER- THEORETIC FUNCTIONS
Transcript of 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, l^i^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
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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^x0^e(log r)V and
at log ?,1-,-Q log X
A log x
" (log2 *)2
/of aj>^o.
(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).
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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
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444 W. J. LEVEQUE [July
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.
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1949] SIZE OF CERTAIN NUMBER-THEORETIC FUNCTIONS 445
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,
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446 W. J. LEVEQUE [July
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
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1949] SIZE OF CERTAIN NUMBER-THEORETIC FUNCTIONS 447
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
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448 W. J. LEVEQUE [July
/ » 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,
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1949] SIZE OF CERTAIN NUMBER-THEORETIC FUNCTIONS 449
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
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450 W. J. LEVEQUE [July
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> —-)}
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1949] SIZE OF CERTAIN NUMBER-THEORETIC FUNCTIONS 451
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 p^i 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)
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452 W. J. LEVEQUE [July
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
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1949] SIZE OF CERTAIN NUMBER-THEORETIC FUNCTIONS 453
(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
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454 W. J. LEVEQUE [July
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
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1949] SIZE OF CERTAIN NUMBER-THEORETIC FUNCTIONS 455
| 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)).
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456 W. J. LEVEQUE [July
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)).
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1949] SIZE OF CERTAIN NUMBER-THEORETIC FUNCTIONS 457
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($(n))1/2.)
This is Lemma 4 of [7].
Lemma 4. Denote by ln the number of positive integers m^n for which simul-
taneously
(12) /„„(«) < /L„ + coi73a„, fan(m +1) <Aan + a,,73a„.
7/se» /„ = nD(ux)D(o)2) +o(n).
Divide the integers ?w=w which satisfy (12) into classes Exx, Exi, Eix,
E13, En, E3x, • • • so that mCEEa if and only if simultaneously
\p(m, n) = a,(»), ip(m + 1, ») = a¡(n),
and denote by \A\ the number of elements of A. Clearly
in = 2Z\Eij\= E I&/1+ E |ä/|.i,¿ <riiO|,a/£Ai >'.j';<i,->jSn or ay>3„
By Lemma 3,
E |£«| < J«(4>(«))"2,Ci>ßn or aj>/S„
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458 W. J. LEVEQUE [July
and therefore it suffices to show that
J2 | Eii\ = nD(o>x)D(u2) + o(n)
as »—» oo.
By Lemma 2,
E |a,l-|ft(i--) E'^p^, . ai.ajäßn 4 4_2 \ g,-/ n,,a,â|S„ ^K^í)
+ .(.ñ(i-2-) £'^^).\ »-2\ g.7 a„aj£ßn <t>(aiOj) /
where the dash indicates a summation over the a,-, a,- satisfying
/«„(«<) < ^4a„ + wi5«„, /<»„(«,) < Aan + co2Ban, (a„ a¡) = 1; 2 ¡ a«!,-,
7J(«i, a/, ») = II ( 1 + —,-rr) •si«ST..S4|a<«,\ Çi(g.- - 2)1
Now divide all the positive integers into classes Fxx, F12, ■ ■ ■ such that
mÇzFij if and only if xp(m, »)=a,(»), xp(m + l, n)=as(n), and let {Fij} be
the density of T",-/. Now consider the set E'T7»;, with the dash as before.
Putting l = an and using Lemma 1 we have that
{E'T7.;} = D(»t)DM + o(l)
as «—► oo.
Now
(14) E'F.-i = E' *« + E' *«.a,-,a,S0n a,->0„ or a/>0„
and by Lemma 3,
(IS) { E' F«l < 6($(»))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;,
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1949] SIZE OF CERTAIN NUMBER-THEORETIC FUNCTIONS 459
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.
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460 W. J. LEVEQUE [July
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* ~ TJ^ii^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
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1949] SIZE OF CERTAIN NUMBER-THEORETIC FUNCTIONS 461
,>-¡/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
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462 W. J. LEVEQUE [July
(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) \ )
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1949] SIZE OF CERTAIN NUMBER-THEORETIC FUNCTIONS 463
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).
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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.
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(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
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11. M. Kac, Note en the distribution of values of the arithmetic function d{m), Bull. Amer.
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Harvard University,
Cambridge, Mass.
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