An Essay on the Distribution of Values of the Ct-Function ' and Related Topics

AN ESSAY ON THE DISTRIBUTION OF VALUES OF THE CT-FUNCTION ' AND RELATED TOPICS

ANITA STRAKER ProQuest Number: 10096394

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The function ofn^crtniis defined to be the sum of a ll positive integer divisors of n • This dissertation is a survey of major work done during the last 30 years on problems connected with numbers n satisfying gia) ^ A, X > 1. n For real values of X these numbers are called X -abundant. When x= z , the numbers are simply called abundant. When A is any integer, but the equality sign only holds, the numbers satisfying the equation are called multi-perfect, or, for the special case x=2 , perfect. There is also some discussion on pairs of amicable numbers C^i^) , for which • cXpn^ = = m-v n . The first chapter contains those standard results of number theory which are essential for the development of this dissertation. The reader is advised to proceed direct to Chapter 11 and to refer to these results as they arise in the te x t. Chapter 11 is an introductory chapter, in which the historical back-ground of these numbers is presented. It includes theorems discussing the general behaviour of the function ^ , for example i t s average value, and true maximum order, and a deeper theorem finding the number of distinct numbers ,for all integers m not exceeding n • The third chapter contains a very full discussion of abundant and x-abundant numbers. Conditions for consecutive abundant numbers are found, the sequences of abundant numbers and X-abundant numbers are sgiown to possess density, and bounds for the numbers of primitive abundant, and primitive X -abundant numbers not exceeding n are determined. Chapter IV concerns perfect and multi-perfect numbers. The density of the sequences of both these types of numbers is shown to be zero, and upper bounds for the numbers of both perfect and m ulti-perfect numbers not exceeding n are found. The final chapter contains a brief discussion of amifcable numbers. Considerably less is known about these, and the chapter contains only one major theorem: namely that the density of the sequence of amitable numbers is zero. The theorems incorporated into this dissertation are mainly those of P.Erdos, whose work in this particular field has been outstanding. 1.

NOTATION.

a b, m,n, denote integers. The letters and will be used ^ without exception to denote primes. denote real numbers. c, c,, Cz, denote absolute constants. Z f(m) ; "U fü>), ( with various modifications and extensionswhich will be explained in the text ) indicate sums and products respectively over all positive integers m , or all primes , within the specified ranges. In the case of the product, where no range is indicated, it is understood that all primes are to be included. o, O, are used in the c la ssic a l sense. If f(n) and are functions of the integral variable n , then ( 1 ) , means that = O (2 ) f- ^OC0 ) , means that ,for all values of n . ( 3 ) [ ^ , means that f /(ÿ = 1 . denote respectively the lesser and greater of a and b denotes the largest integer which does not exceed X- . b|a, bfcL, mean that b divides a , or b does not divide , respectively. a= b means that m divides a-b • (OL,b) - means that d is the highest common factor of 1 . (In our case s will only take real values). Sn, . denote respectively the square-free and quadratic parts of an integer m . i.e. if the prime decomposition of m is , where each oil > % , then ^ and Gn'- 2.

CHAPTER 1.

PRELIMINARIES.

1 . 1 . This chapter contains a number of auxiliary results which will be used ^n the sequel. These fall, broadly speaking, into three main groups: one covering standard results on the distribution of primes, another giving some simple relations between an integer m and its square-free and quadratic parts, and a third defining some number-theoretical functions and proving any necessary re su lts concerning them. We list these results as a number of lemmas, (though some, of course, are major theorems in their own field), and give the proof whenever i t is short. In the case of non- elementary results we give a reference only. The ordinary analytic theory of natural logarithms and exponentials is taken for granted, but it is important to lay stress on one property of log X, . Since

> r ! ( r + i ) ! ^ cxs DC XA , I) ’

Hence, more rapidly than any power of x . I t follows that Icgx , the inverse function, tends to infinity more slowly than any positive power of x • , but

JC^ for every positive 5 • Sim ilarly, loglcngxL-?w more slowly than any power of logx At this stage, however, the reader is advised to proceed direct to Chapter H, and to refer to the results of this chapter as they arise in the text.

1.2. In this section we collect together some standard results on the distribution of primes.

LEÏ4MA 1.

— m m 5 jc- where ^ Is a constant, known as Euler* s constant.

Proof. We have ^ = I + ^ - ^ , ) msjc If ms W - I I ^ _ cut

1 f r'-" M dk VI \ + f ^ W air — r [Ü dfc Jl J&C]

I ^ logx t J" dt - -±) 4. log X -I- fp^J — b__C^3

Icrgx +- ÿ + , where

r" h - DÜ dt , 4. — [ T - is independent of x , and

r" t - W . c^ 3 I ^S^QWat*- oc-k) = oc±) Jx ^ 3 T fc. which completes the proof of Lemma 1 .

LEMMA 2.

rr Cl diverges to zero.

Proof. Consider the finite product

- » n c i - i r ) b ' N K) 0 (W) m = I m where f i, if each prime factor of m does not exceed N, otherwise . Therefore,by Lemma 1 ,

-I m hn-t rH Hence n ( 1- ^ ) ' ' -4, lA, os N w , \>^ IS and Lemma 2 follows immediately. 5.

LEMMA 3 . ZI cüvtrges ], Converges

Proof. The first part follows from Lemma 2 , and the second part by comparison with the convergent series

LEMMA k.

LEMMA 5."*^ X = leg log o + B -t- '

where B is constant.

LEMMA 6 .

TT C'-p-) ^

where ^ is E d ler's constant.

LEMMA 7 . c," logn > T - r c '^ - ^ ) >

for suitable constants cGoand c">o.

Proof.

and the result follows from Lemmas 3 and 6 .

^ E. A. INGHAM, D istribution of Prime Numbersj' Cambridge Tract No. 3 0 , Theorem 7. 6.

LEMMA 8 . ^■2 ileglogn.

n Proof* The^ le f t hand sum may be w ritten as ; oC^ 2.

F ^ k F o

__ I + pen ^ ^smen '«("’'O

- pfnt- ^ ^

•< .2 logl(jgn, by Lemma 5 for all sufficiently large n ,

LEMMA 9* n t" < 4-". pan Proof. Consider the product ^ . It is clearly a factor of , and it is easily proved by induction th at

r r ^ - 4 -^ »

We now prove Lemma 9 by induction on n . The inequality is clearly true for n = z,:s . If n is even, then 7.

TT b * TT b . jj£ n W- n-'

Hence, we need only prove the result for n odd. Put n = 1 . Then by the induction hypothesis

TT b - TT b . TT f>

- f ^ and th is completes the proof of Lemma

LEMMA 10.*' (Tchebycheff) If TT(nj denotes the number of primes not exceeding n , then there exist two absolute constants C,>oand Ci >0 such that

^i -G- ^ T T ( j\ ) ^ O n (ogn lugn

LEI4MA 1 1 .*^ (Prime Number Theorem).

for any fixed (positive) > l .

LEMMA 1 2 .“(A lternative form of Prime Number Theorem)

lim -J- .2. log p = J- . W ^ n

E. A. INGHAM. lo c .c it. Theorem I+. ('îbé. nghrhana. incgiiaiitg ét^JvaVértat to our LÉHnnma. 4^ % E. A. INGHAM. lo c .c it. Theorem 2]. ^3 E. A. INGHAM. lo c .c it. Theorem 3 . The result then follows from Lemma 1 1 . 8 .

1 . 3 . We now prove some simple relations between an integer m , its square-free part -Sm , and its quadratic part Gn . We bave already shown that if the prime decomposition of m is t^en Sfvi - 1^1 pz. j Grn ~ ......

Trivially, we have the following:

LEMMA 13. C ^ Cfn ) - 1 .

We now prove two simple results about Gn

LEMMA Ik. Fm caay always be written as the product of a square and a cube.

Proof. Every integer greater than or equal to 2 , (and hence every ), may be written in the form sb for some integers ol and b , where cLa o and b $. o . The proof of Lemma Ik now follows immediately.

LEiMMA 1 5 . Gn is always divisible by a square not less than 9.

Proof. By Lemma we may write r„ - where b is square-free, and where for every f» | b we have

Hence

b & CL ^

Tm *

i .e . cl" ) -

Finally, let denote the number of integers m ^ n for which Gr, > R , where A is a positive constant. We prove

LBlviMA 1 6 . RCn, f\) ^ c n R'^% where c is a positive absolute constant.

Proof. By Lemma 15,

‘ r ? ,n t

‘ " 5, -P- ^ R ^yz.

n R ^ ^ l'vx 1=1 10.

where c ^ O is an absolute constant.

l.k. In this last section we consider some properties of some of the better known arithmetic functions. An arithmetic function is a ^eal function defined on the natural numbers, which describes some special property of these numbers We define first the divisor function 'rcn) as the number of divisors of n . We shall show that is multiplicative, i.e. ( 1 . 1 ) TCma; = if (m, n) - 1 . . Suppose th at the prime decomposition of n is

n = h " ' ...... h / % Then a typical divisor of n has the form

h ...... where 0 ^ • The total number of divisors of n is then the number of distinct sets of exponents y with

O 6 ^ . Hence, (1 . 2 ) XCn) . 0 ..... and ( 1 . 1 ) follows from ( 1 . 2 ).

We prove also two simple results about the behaviour of r(n) as a function of n . Firsts 11.

LEMMA 17 . (Dirichlet). The average value of T(n) is )crgn .

Proof. We denote by D the region in the upper right-hand quadrant contained beti/^een the axes and the rectangular hyperbola

We count the lattice points in D, including those on the hyperbola, but not those on the axes. Every lattice point in D appears on a hyperbola ^ ^ j ( I ^ 5 ^ n and the number on such a hyperbola is r(s^ . Hence, the number of lattice points in D is ^

Z + T(z) +— f r(n 1 , m s n

Of these points, n - CnJ have the x-co-ordinate 1, have the x - co-ordinate 2 , and so on. Hence, by Lemma 1 , th e ir number is

^ t C^) = [nj + [ -g;] + L-^J ^ n ^ 1+^ + ...+ ^ )+ nt t rj

n (crgn -f and th is completes the proof of Lemma I 7 . Secondly, we find an upper bound for the order of 12. magnitude of -c(n)

LEMMA 18.

T t n ) = for ail positive 6 .

Proof. The assertions that ir(n] = , and T(n) = O(n^) , for ail positive 6 , are equivalent, since when 0 < 6' ^ S . Therefore, let the prime decomposition of n be n = |>r ..... By (1.2), (1.3) ^ /

Since w We have

Now i f jp > 2^ ^ , we have oi-t 1 <: fi±_L 1-^ 2 . ^ Therefore, by (1.3) and (1.H-),

and th is proves Lemma 18 .

The second arithmetic function we consider is the 13.

Moebius function ^(n) , where

i , i f n ^:l ,

4 O, if n is divisible by a square, (-1)*^ if n is square-free and bas r d is tin c t prime fa c to rs. It is clear from this definition that [^(p) is also multipli cativ e. We shall prove three simple lemmas about this function.

LEMMA 19.

1 , i f MW ct|n O, otherwise

Proof. The result is obvious for n = 1 . Suppose o ? i and th at the prime decomposition of n is

By the definition of /u(n) all we need consider is the sum

|*r = (14- ( - 1 ) )

= o ,

LEMMA 20.

/U ( a ) a?-|rr<

Proof. Let b be the largest square dividing m . Then Ilf.

2 : f^ (a ) -- ^ {*.(0^).

But, by Lemma 19> the sum on the right-hand side is 1 i f b ^ l ^ (i.e. m is square-free), and is 0 otherwise. Our lemma then follows by the definition of ( ^(m) ( .

LEi^iMA 21 . Let (1.5) F (n , 0 = S I I . rt (rn,rj = 1_

Then

F/n.O " h" + 0 (n'/"TCrJ)) rr^ Mr

Proof. By Lemma 20 ,

m s n CL (a,r)-'X (b,r;=a.

^(CX) ( ? ( ^ upi (a ,r ) z 1

where, by Lemma 19,

G(x,r) = ^ s ^ bijt d-ICM'-) d.|b 15.

Hence ^

FC n,r) ) s . d \i / CK 6 Jn 0.2-

ê . a. = 1 a ? '

But, by Lemma 3 , IT( i- converges. Therefore ^

= nc ^ ) " = rr( +

SO that

‘"Cn,r) = ^ g ( '+ + O(nV^rW).

The last arithmetic function that we consider here is v(2n) , where v(n) is the number of d istin c t prime divisors of n • i*e.

^ J- ; ''CV = O .

We note that v(n) is an additive, not a multiplicative, function: s/C^m) zz v(n) vc«vi) , )p (n , Kv\) - ]_ , We now prove one la s t lemma:

LEMMA 22. 16.

v(n) s , !<^ g lcrg z_

Proof. Let the prime decomposition of n be

Then

VC^I) - r 6 159-0. • I0 9 Z

This now completes the list of the auxiliary results which it will be necessary to use in the following chapters. 17

CHAPTER 11.

INTRODUCTION.

2.1. One of the oldest and best known of all the arithmetic functions is cr(n) , the sum of all the^divisors of n . The earliest reference to it seems to have been by the Greeks, who were aware of its multiplicative property: (2.1) cr(mn) = ^ »f- Cnn,n) = 1_

The proof of ( 2 . 1 ) is very simple. Suppose the prime decomposition of n is n - A typical divisor of n then has the form where , Therefore, the sum of all the divisors of n is given by

and ( 2 . 1 ) follows immediately. 18.

n and. ( (xjct ctiv/^sov’ n alwizoj^ Since n—d ivid es—tts e lf-, we c le a r ly have

LEMMA 22. q£n2 >■ J- , " > 1 n

This raises the interesting question of when (2. 2)

and X >1 . Strangely enough, although th is question seems simple, it has lead to problems of great difficulty, some of which remain unsolved to-day. In this essay we attempt to answer this particular question, and some others closely related to it, by giving a collected account of all the work done in this field, mainly during the last thirty years. It is important to emphasise the inspiration which this work has derived from experiment, which takes the form of testing possible general theorems by numerical exanples. Such experiment, though necessary in some form to progress in every part of mathematics, has played in . a greater part the development of the theory of numbers than elsewhere, for in other branches of mathematics the evidence found in this way is too often fragmentary and misleading.

2.2. We consider first the case when (2.3) - 2 . A number n for which (2.3) holds is said to be perfect. This name originated from the Greeks, who thought of a perfect number as one which equalled the sum of its proper divisors. 19.

Since the days of antiquity perfect numbers have been essential, elements in all numerological speculations. God created the world in 6 days, a perfect number. The moon circles the earth in 28 days, again a symbol of p erfection.

Only one general class of perfect numbers is known. In the ninth book of Euclid's Elements we can find the follow ing Theorem.

THEOREM 1 , (Euclid). If 2"^' - ) is prime, then is perfect.

Proof. Write 2 '^^’ — I = rn - I'^^^then

cr(nn; - ])(jp^ l) = I) - 2rv7, SO that rv| is perfect.

Theorem 1 shows th at to every Mersenne prime (i.e. one of the form 2^-1 , where p is prime), there corresponds a perfect number. It is not yet known whether there are infinitely many perfect numbers of this form, since it is not known whether there are infinitely many Mer senne primes, but in the eighteenth century Euler succeeded in of- proving th at every even perfect number must be^Euclid's form.

THEOREM 2. (E uler). Any even perfect number is of the form 2" , where 2"^^' — I is prime. 20.

Proof. We can write any even perfect number in the m = jz.^b form Ij , where n > o and b is odd. Hence,

n+l - C-(b) 2*"^'

The fraction on the right-hand side of this equation is in its lowest terms, and therefore b = &Ch) = where (X is an in teger. Now i f cxy 1 , b at least has the divisors a, t>, 1 , so th at axp) ^ a+b+l = l'^'a+I > an evident contradiction. Hence a ^ 1 , and we have

m = 0 , and - i) ^

But, if 2^^' - 1 is not prime, it has divisors other than itself and 1, and

Hence^ 2^'*'* — | is prime, and the theorem is proved.

In Barlow's Number Theory (London, 1811) the author gives the perfect numbers up to the one corresponding to M , at the time the g reatest prime known. This perfect number 21.

'is the greatest that will ever be discovered, for, as they are merely curious without being useful, it is not likely that any persoh will attempt to find one beyond it*. The great efforts expended since that time in such in-± 5ug^ computations show how easy it is to underestimate human curiosityl In 1876 Lucas ‘found a method fo r te stin g whether or not a p a rtic u la r integer is a Mersenne prime, and used i t to v erify the prim ality of 2.’^^ —1 . This remained the largest known prime until recently, when the electronic computers developed during the second world war became available for peaceful " purposes. In 1956 the S.W.A.C. computer at Los Angeles determined all primes of the form 2 ^ - I for p , giving seventeen Mersenne primes and hence seventeen known perfect numbers. Since — I is a number of 925 digits this gives some idea of the rarity of the even perfect numbers. The first five are 6 , 2 8 , 4 9 6, 8 , 1 28 , 3 3 , 550, 3 36. T\i6 éxièttncé The question of^odd perfect numbers is one of the celebrated unsolved problems in number theory. I t seems probable that there are no odd perfect numbers, since extensive numerical calculations have failed to find one less than e or with less than 2,800 different prime fa c to rs.

L. E. DICKSON. History of the Theory of Numbers. Vo}. 1. Ch. XVll. MITSUI, Sci. Papers. Coll. Gen. Ed. Unif. Tokyo. 6 (1956) 1 - 1 1 . 22.

It has been possible to find various criteria which any odd perfect number must s a tis fy , and the two cla ssic results in this direction cam from Euler and Sylvester. The la t t e r showed that an odd perfect number must have at le a st five distinct prime factors. We do not prove this result, since it is interesting rather than useful, but we shall prove Euler's result in our next theorem.

THEOREM 3 . (E uler). Every odd perfect number is of the form

m = q. = oC a I mmXT.

Proof. Let

m ...... where q,,, .....are distinct (odd) primes. Since m is perfect

crcm ) =

Thus, one of the numbers ...... , say j is the double of an odd number, and theremaining ones are odd. But (2.4") ... and each of the <

^ SYLVESTER. Comptes Rendus P aris. 106 (1888) 448 - 4^0. 23. the right-hand side of (2.40 and is the double of an even number. Sim ilarly, i f q,, is of the form 4i-+5 , then ^ I # But, since crccv,‘*9 = i + q^i + it follows that 4 | crC^T') , and is the double of an even number. These contradictions complete the proof of Theorem 3 .

In recent times, the most notable work on the question of odd perfect numbers na = has been by Kanold, who in a series of papers , eliminates possible forms of 3k , by Touchard who shows th at m must take one of the forms

3 br+ I , 5>br4- Q ^ 3 b r + 1 3 ^ 5 br+ '3k and by McCarthy ^ , who proves th at m ^ Zmoxis, and i f m a I , then oC ^ -1 rvicrtl3 ,

2. 3. Returning again to the question (2.2) we shall more generally consider the case when

H. J. KANOLD. J. Reine. Agnew Math. I 9 I+1 ,1 9 ^ 2 , 1 9 V+, 1 9 ^ 0 , 1 9 ^3 . J. TOUCHARD. Scripta Math. 19 (1953) 35 - 39.

* 3 P. McCarthy. Amer. Math. Monthly. 64 (1957) 2?7 - 2^8. 24.

(2.5) ^ and A is an in teg er. A number o for which (2.5) holds is said to be multi-perfect, and when the value of X is particularly relevant is said to be of class A . The reason for this nomenclature is apparent. Clearly the perfect numbers are a sub-class of all the multi-perfect numbers. The problem of finding multi-perfect numbers appears to have been formulated f i r s t in I 63I by Mersenne in a l e tte r to D escartes. The l a tt e r must have speculated con siderably over the proposed problem, because seven years later he responded with a list of multi-perfect numbers together with various general methods for finding them. The first two of these were the multi-perfect numbers of Class 3 ^ izo = 2^ 3 .3 and 672 = 2^. 3.7 . Fermat and Frenicle also worked on the problem, and letters exchanged between them contain several other multi-perfect numbers. More recently many more have been discovered, notably by Lucas, Lehmer, Cunningham, Carmichael and Mason. The standard list of all known multi-perfect numbers was published by Poulet in 1929, and revised in 1934. This contained 334 numbers, some of class as high as 7 . New discoveries since that date were tabulated by Brown in 1954.

P. POULET. La Chasse aux Nombres. 2 Vols - Brussels 1929 - 1934. ^ A. L. BROWN. Multi-perfect numbers. Scripta Math. 20 (1954) 103 - 1 06. 25.

2.4. For numbers that are not perfect there are two possibilities :

( 2 . 6 ) ^ ; or , n n

Numbers of the first kind are called abundant, and those of the second kind deficient. Generally speaking, we class the perfect numbers with the abundant numbers, so that an abundant number is one fo r which

(2.7) n A number n satisfying our general equation (2.2) will be called X-abundant. The distinction between abundant and deficient numbers has always been considered important in numerology. For instance, Alcuin (735-804), the advisor and teacher of Charlemagne, observes that the entire human race descends from the 8 souls in Noah* s Ark. Since 8 is a deficient number, he concludes that this second creation was imperfect in comparison with the first, which was based on the principle of the perfect number 6 i The first few abundant numbers, all found by experiment, are, 6 , 1 2 , 1 8 , 2 0 , 24, 2 8 , 3 0 , 3 6 , ...... ) there are only 23 of them up to 100 , as the reader may easily v erify , and a ll of them are even. The f i r s t odd abundant number is ^ 4-6 = 26.

The very earliest mention of these numbers appears to have been by Nicomachus in A.D. 100, who cited 12 and 24 as abundant, and 8 and l4 as d e fic ie n t. In 1296 Jordanus Nemoranus proved:

THEOREM 4. A prime or a power of a prime is deficient.

Proof. The proof is very simple.

- I = I + p ^-\ ^ I 4 -L- ^ 2. ^ and our theorem is proved.

We can also prove ( c .f . Theorem 1 ): THEOREM 5. ) is abundant, for all n ^ 1

Proof. Since, for all values of ol , CT(CX) ^ I -I- it follows that

Hence,

Therefore, 0 ) ^ z. . 2P (J2."+'- I ) 27.

The following lemma, has enabled us to prove some more simple Theorems about abundant numbers.

LEMMA 2 3 . If d|n , then

^ °UÙ ■

Proof. Let the prime decomposition of n be

Since cn(n) is multiplicative, we need only show that if O 6 p s , then

P~( < a-Ç ,

i.e. r ' which is clearly true.

It follows immediately from Lemma 2.3* that

THEOREM 6. (1) A m ultiple of an abundant number is also abundant. (2) A d iv iso r of a d eficien t number is also

d e fic ie n t.

Theorem 6 raises another interesting question. It is clearly possible to find a minimal sub-set S of the abundant 2 8 . numbers, so that the proper divisors of each element of S are deficient. The ^âaset# of all the abundant numbers is then the set formed by all multiples of the elements of S. L. E. Dickson^ was the first to see the possibilities that this question evoked, and in 1913 he defined a primitive abundant number as one fo r which

> Z . ) rvi but for every proper divisor d m ,

Dickson* s work included a number of minor r e s u lts . He proved that there is only a finite number of primitive abundant numbers having a given number of d is tin c t odd prime fa c to rs, and a given number of fa c to rs 2 . He showed also th a t there is no odd abundant number with fewer than 3 distinct prime factors, and gave a list of the numerous primitive odd abundant numbers with four distinct prime factors, as well as lists of even abundant numbers of certain kinds. In particular he determined all primitive abundant numbers less than 1 5 , 0 0 0 . Extending the idea of primitive abundant numbers, we shall discuss integers n for which ( 2 . 2 ) holds generally. We shall say that n is A-abundant if ( 2 . 8 ) ÇïnJ ^ X ;> 1 , n ^

^ L. E. DICKSON. Amer. Jour, Math, 35 (1913) 4-13 - 426. 29 and n is primitive X-abundant if (2.8) bolds, but for every proper divisor dL of n ,

X . oL

2 . 5 * We have already introduced the notion of a perfect number, and explained that the name originated from the Greek terminology where

cr, (

Pairs of numbers which satisfy ( 2 . 8 ) are said to be amicable, and numbers of this type have been even more prominent in the lore of number mysticism than the perfect numbers, having symbolized mutual harmony, perfect friendship, and love. Their existence seems to have been discovered somewhat la te r than the perfect numbers, probably in the period of the flowering of the Neo-Platonic mystical school in Greek philosophy. One of the most influential of the Neo-Platonic philosophers, lamblichus of Chalcis (about A.D, 320), ascribes the knowledge of amicable numbers to the earliest Pythagorean school, about 500 B.C. In Arab mathematical writings, the ami cable numbers occur repeatedly. Th^play a role in magic and astrology, in the casting of horoscopes, in sorcery, in the concoction of love 30.

potions, and in the making of talismans. As an illustration we quote from the Historical Prolegomenon of the Arab scholar Ibn Khaldun (1332 - l406). 'Let us mention that the practice of the art of talismans has also made us recognise the marvellous virtues of amicable numbers. These numbers are 220 and 284 One prepares a horoscope theme for each individual On each one of these themes one inscribes one of the numbers just indicated but giving the strongest number to the person whose friendship one wishes to gain, the beloved person. I don't know if, by the strongest number one wishes to designate the greatest one or the one which has the greatest number of aliquot parts. There results a bond so close between the two persons that they cannot be separated*. Through the Arabs knowledge of amicable numbers spread to Western Europe. They are mentioned in the works of many prominent mathematical writers about A.D. 150 0, fo r instance Chuquet, Stiefel, Cardanus and Tartaglia. However, there is no indication of any other pair of amicable numbers (besides the pair ( 2 2 0 , 284) known to the Greeks and the Arabs) having been discovered before the work of Fermat. This is somewhat odd in that Fermat found his new pair through the rediscovery of a rule that actually had been formulated by the Arab mathematician Abu-l-Hasan Thabit ben Korrah as early as the ninth century: 31. i f |3 = ^x3.2"-'-1 , and. r = 9..2"' - I , are all primes, then the numbers a. = 2"^ , b = 2''r_, are an amicable pair. (It is easy to verify, with this formulation, that cr^a^ = o-^b) =a-f b ). The case n - 2 , gives l3 = II, q, - S, r = 17^ and we obtain the classical pair ( 2 20, 284). The next two pairs found by this rule come from the value n = -f- , and

n 7 , giving the amicable numbers

r II, Z9(b290 = = Zt Z t 23.47.Z3.47. j [ 9,365,9,363,684- - Zl Ml. 383. L 18, 4*6 - Z-^. 1151. L 9 ,4 3 7 / 0 5 0 - Z^. 73Z7.

Euler took up the search for amicable numbers in a systematic fashion, and developed several methods for finding them. In 1747 he gave a list of 30 pairs, which he later expanded to more than 6 0 , all of them exceedingly large. However, it is interesting to see how the purely experimental side of number theory can defeat the ablest of mathematicians. More than a hundred years later, a sixteen year-old Italian boy, Nicolo Paganini, published the very small pair

184 = Z^. 37 , (Z/O -2 .5 .1 1 ^ , which had eluded all previous investigators. There is no evidence as to a method of discovery: they were probably found 32.

by t r i a l and erro r. A complete survey of the existing knowledge about amicable numbers was published in 1946 by E. B. Escott . This paper contains a list of the 39O known pairs in factor form, together with a discussion on various methods of discovery.

2 . 6 . Since, in this essay, we are to be concerned with numbers arising from the function (in the case of perfect and abundant numbers), and from the function (in the case of amicable numbers), it is interesting, at this stage, to consider the average value of these functions. We prove

THEOREM 7 . The average value of I's n ^

It 15 ecLsy -Vo |>rnw 'V n a i Proof. Ab on easy- applioati-eB- of multiplication of Dirichlat sarieo involving the Riemann-zota funcMon--^(aj-we have

a , '

where— , P utting 5 ^ 0 into thi-e- equation-y-ond restricting the summatien-to the integers— n —, gives-

if- E. B. ESCOTT. Amicable numbers. S crip ta Math 12 (1946) 61 - 72 . 33.

But

and by Lemma 1, ( 2 . 1 2 ) r^K, " ^ C '«3 n ;.

T herefore, 2

Since — I & , this suggests that there are far more deficient numbers than abundant. This is certainly ’

true of the integers less than 1 0 0 0 . Only 22 of these are abundant. The equivalent Theorem for the function is very similar :

THEOREM 8 . ^ a - (m ) = O(^o/o-gn)- mS:K7

Proof. mSrH where the summation extends over all the lattice points in the region D of Lemma 17. Hence,

^ ± L f J C L f J ^

n 3 ^-

Then, by (2.11) and (2.12), otm) = + 0(nlcrqV7j, Win n In particular, since JZ n /\/ , we have the average value rvini of is -^TT^ n .

The erro r terms in Theorems 7 and 8 are, of course, very far from being the best possible. We can re-arrange ( 2 . 1 0 ) and ( 2 . 1 1 ) so th a t a. ^ " A. X. - -PM

and ^ crCmJ = -k-TT^n^ - np^n) ^ n\srn where

^ m C( -% rn - C-^3 -

Wigert showed as early as 1913» th at y O (n ) ^ 0 ( l o g n ) , w W idifoll(>>^s,ofcaruoe, jroi»\'nri<-abcwak)sit.

Waifisz improved on this estimate of Wigert, by the use of Weyl's in eq u ality for exponential sums, to

p/n) - o( -im — ) . ^ ^ IcralcrzinIcrgiogn J

Further progress, however, became possible after Vinogradov's

S. WIGERT. Sur quelques fonctions arithmétiques. ActaMathematica 37 (1913) 113 - 123. A. WALFISZ. Tellerproblem e. Math Z e its. 26 (1927) 66 - 8 8 . 35. remarkable improvements on Weyl's inequality, and Davenport and Walfisz proved* independently that for any e>o , pen) - 0 ( . By using an improved form of Vinogradov's inequalities for trigonom etrical sums, a Chinese mathematician Pan Cheng Tung claims he can prove by Davenport's method, that c may be replaced by , and this is the best result available at present.

We have shown that the average value of the function is , but the maximum values of the function are considerably larger. We prove another result of Wigert. ^

THEOREM 9 . (Wigert).

nicrglcrgn where ^ is Euler's constant.

Proof. Let the prime decomposition of n be so th at r . I I- U n 3 \ = TT\ I i-I—K ------~ -^ <^ TT » I I___L_ •133 n o ' ' pL

H. DAVENPORT, A divisor problem. Quart.Journ. of Math. (Oxford) ( 1949) 37 - 44.

%2. A. WALFISZ. Uber Gitterpunkte in mehrdimensionalen Ellipsoiden, achte Abhandlung. Travaux de I 'I n s t . Math de T b lis s i. 5 (1938) I 8 I - ( 9 6. S. WIGERT. loc. c i t . 36.

By Lemma 6 , having chosen arbitrarily small, we have th a t (2.14) > O z l l i e l f , Icygïo-gn for all sufficiently large n • Therefore, given any e > o ^ we choose c. n - e and it then follows from ( 2 . 1 3 ) and (2.14) that, for all sufficiently large n > 1 a positive integer to be fixed later. Put ( 2 . 1 6 ) a = ^ so th a t

By Lemma 12, we have th at liw Æ log)) = 3. N-SW |>SN SO that, if N is large enough, (2.17) ^ tog n < , and ( 2 . 1 8 ) IffgH log X log log n -=• log N - 1- log 2.k . Now, by ( 2 . 1 6 )

(2.19) EÜ!) = TT ' ~ . n psN , _

But, by Lemma 3, TT( 1- J converges. Therefore

rrc I - J - 3” - 37. so that, given any e> o , for all sufficiently large H we have

Also, by Lemma 6 , for all sufficiently large N ,

XT' > ('-$)

Therefore, by ( 2 . 1 9) and (2.18),

(2 .Z0 9 (T(n) f I- fr e IcrgH ^ sCki-O

> ( I - fr ) _ / lenlogn - lag ^ 5 ( k + l )

We now choose k large enough to ensure that

— e

Then, if tS is sufficiently large, we have from (2.18)

Icrq %k_ < I c g a k ^ | _ / 1 - 6 \ Iffglogn logN f log ^ & which, after rearrangement, gives,

^^Icrglogn — logilc) > (^ 1—

Hence, for all sufficiently large values of N^and hence for an infinite number of values of n , we have from (- 2..Z0 9 , > (^1 - a ) e ^ Icsglcrgn , and our Theorem is proved. 38 .

2.7. Finally, since we are to be primarily concerned with numbers of the form (T(n) it is interesting, at this stage, to consider the number of distinct values taken by the function , with, We prove a result of Erdos , TvUsk whose work in our particular field has been outstanding.

THEOREM 10, (E rdos). The number of distinct numbers of the form . I s m sr n nn ' ; equals qn + o(n) , where < c, s i .

Proof. To prove this Theorem we shall require the results of two supplementary lemmas.

LEMMA 24. o-(b.) i

Proof. The proof is very simple ; we can clearly assume th at (_b, ,bi) = J . Let the prime decomposition of b, be

b| = ^1 pi- .- f SO th at

Then f>rt«'Cbi) , since [3p-|- 1+ (sp » and all the other factors

^ P. ERDOS. Remarks on Number Theory 11. Acta A rithm etica V (1959) 171 - 177. 38. of slb,) are less than \>r . Hence the equation

LEMMA 25. Let a, , and Oz., be two in teg ers each of whose prime factors occurs with an exponent greater than 1 , ( i.e . whose square-free p art is Ij. Then there exists at most one p a ir of square-free integers b, and bj. satisfying

err a, b j = cr(Q2.b^J ^ ( 2 . 20 )

(Qi,bi_) % bi9 ' (-b|/ bj.) : J. .

Proof. The proof is rather similar to the proof of Lemma 24. We suppose, on the contrary, that there is a second pair of square-free integers b,' and bJ satisfying (2.20). Then we should have, by ( 2 . 2 0 ), th a t

(P PIN qlii) = ctb.'J b,,' ^ t>, e-cb^) b,'

Assume th a t b,,bj.,b(',bi', are so lu tio n s of (2.21) 39. and ( 2 . 2 2 ) for which the product ( 2 . 2 3 ) b, k b,' bz' is minimal. (This product is clearly greater than 1 , since we are assuming that not all the b's equal 1). Let |>r be the greatest prime factor of the product ( 2 . 2 3 ), say (2.24) I b, ; (>rbi.. We can apply an argument similar to that used in Lemma 24 to show th a t t • S im ilarly , f>r + c"Cbz'3 , since the greatest prime factor of b^,' does not exceed f>r . Hence, by ( 2 . 2 1 ) and ( 2 . 2 2 ) (2.25) W I b,' , f b;:'. But then, by (2.24) and (2.25)

also satisfy ( 2 . 2 1 ) and ( 2 . 2 2 ), which contradicts the minimality of the product (2.23). This completes the proof of Lemma 25.

We are now in a position to continue with the proof of Theorem 10. Let I =a,«^az.<...... be the infinite sequence of integers whose square-free part is 1 , and denote by mT’< the in f in ite sequence of in te g e rs whose ■ quadratic part is ai . Put where is square-free, and denote by Si the set of all the numbers 40.

(2.26)

Clearly, no two elements of the set Sl are the same, fo r i f th is were the case we should have

bK") and by Lemma 24 this is impossible. If an element of Si is the same as an element of Sj , J< i- , then we should have

for some and . Let C b/'), = b, so th a t ( 2 . 2 8 ) rnk"mJ" =. aioo-k', aibd-k"', -- 4 , d,^T)--1.

Then, if (2.27) holds, wo should have

. . C-(a: dk"') = o-(a, d«X” )--1 .

and by Lemma 25 there is at most one pair of square-free

integers d«"’ and such that (2 . 2 9) holds, with J*^i

Therefore, there are at most l-I of the integers du''-* , for fixed i, and we denote these by ( 2 . 3 0 ) dv,‘‘\ d * /'), ...... d K , and their product by Di = d

It follows that, if (2.27) holds for some , then rMn"-* must be divisible by at least one of the integers ( 2 . 3 0 ) and h i . hence have a factor in common with We now restrict attention to those integers which do not exceed n . We denote by Nc the number of elements in St which differ from all the elements of the classes Sy , J ^ L . Then Me certainly does not exceed the total number of elements of Sc , and is at least as great as the number of integers not exceeding n which are co-prime with Dc . Therefore,

N t /^(W| ( 2 . 3 1 ) bs- -0ac_ (byaO^l (b;aO-J- Lb) DcO = 1

By Lemma 21 and Lemma 18, the sum on the right-hand side of ( 2 . 3 1 ) equals

'= (* .“0 - ^ ° C S r ) -

Similarly, the sum on the left-hand side equals

n^ac ^ ^ cxi J

It follows that

( 2 . 3 2 ) N c -- t where

A o T T ^ T T ( 2 . 3 3 ) ■O.TI' and to prove our Theorem we must evaluate the sums over all integers L ? o , of the expressions on the right and left-hand i sides of ( 2 .^3 ). 42.

Now we bave

(2.34) f 4^t. = TTCl^ ^ ' V^ P(t>-0sk-o ) •

Also, by Lemma 21, the density of integers whose quadratic p art is cxi equals U/w, = -fe- TT C' + -k3"'. p; TT^CXl ÿom 5im^e contiderrxWnô It follows/1 hat

U .3 5 ) J

In addition, since a, ^ 1 , and D, = 1 ,

( 2 . 3 6 ) Ji^cOc ‘^' ^ P ' " * É . > -fc_ . TT^

It now follows from (2.32), (2.33), (2.34), (2.35) and ( 2 . 3 6 ), th a t

Mi = C,n + o ( n ) , L= I where ^ t c, s 1 , and this completes the proof of our Theorem. ^3.

CHAPTER 1 1 1 .

ABUNDANT NUMBERS.

3 . 1 . We have already defined an abundant number n in 2."t as one for which ( 3 . 1 ) ^ > 2 ., and a primitive abundant number n as one which satisfies ( 3 .I) iS^iuViTkol' but^for every proper divisor d of n , ^ X. Over the last thirty years Erdos has succeeded by entirely elementary means in giving an astonishingly complete account of the distribution of these numbers. We shall prove below that there exist arbitrarily long runs of consecutive abundant numbers, we shall find the correct order of magnitude for the counting number of the sequence of primitive abundant number, and we shall show that the asymptotic density of the sequence of abundant numbers exists.

3 . 2 . The most amazing part of Erdos's result that there exist arbitrarily long runs of consecutive abundant numbers , is that at the time of publication of his paper there was no empirical evidence of any kind to suggest that such is the case

^ P. ERDOS J . London Math. Soc. 10 (1935). 128 - I 3I. 44.

Glaistier's number divisor tables for all the integers up to 10,000 were only published in 1940, but even these are not helpful in this respect, for there are only eleven odd abundant numbers amongst them, (the smsillest being 945), and they do not lieclose together. The only possible numerical hint that could provideany motivation for looking for such a r e s u lt, comes from Salie ', who showed in 1955 th a t every integer greater than

33,426,748,355 is either an abundant number or the sum of two such numbers. We must remember, though, that Erdos proved his theorem some thirty years earlier. To prove this theorem we shall require the follow ing two lemmas:

LEMMA 26 Let M,, ndi,---Mr be r in teg ers each of which is less than n . Then

vC Mr ) < TT(4 r for all sufficiently large n .

Proof. We have already shown that v(m) is an additive function, so that it follows £rom -Lomma-22 th a t

H. SALIE Math. Nachr. l4. (1955). 39 - 46. The mfennAlih) vO*'") \i(.n\)+v(ii) is+TV\AAUy+rwfc. ^5.

(3.2) v(Lm,nn^.-.-nOr ) ^ Dogn . lagZ

But, by Tcbebycheff*s Theorem (Lemma 1 0 ),

(3*3) TTC^-rlcg^n)> 5r Icrq^n _ > r bgn ^ (r^n) > log L4rlog^n9 lugZ providing r? is sufficiently large. ( 3 . 2 ) and ( 3 . 3 ) complete the proof of Lemma 2 6 .

LEMMA 2 7 . Denote by — rwr the integers between fn-N-f-1 and m which are not divisible by any prime less than or equal to a given prime cy . Then

'' » % ond rvi arz ^ ^ if H «• sufficiently large.

Proof. Let TT "b = Gi. . Then

r = Z."" Z. , cx.= m-Hi-i dlo. d\ci. since, by Lemma 19, the inner sum equals 1 when (a ,o .) = L , and .is 0 otherwise. Reversing the order of summation we obtain

r -- [ & ] -

N Z ^ o ( ’=CQ.)) dia <=*•

> tL ^ ,

' -ïTî-^'-p)) 46. i f M and m axe sufficiently large compared with .

We are now in a position to prove the following r e s u lt.

THEOREM 11 (Erdos). It is possible to find two positive constants c, and Cj. such that for all sufficiently large n there exist at le a s t c, log log leg n , but not more than tog log log n ^ consecutive integers all abundant and less than n .

^roof. To prove the first part of this theorem it will -e n ly be—necas3ary-to show that there is a sequence of M abundant numbers, ...... -- , say, which are relatively prime in pairs, and such that

(3*^) m.M,. c n . For in this case the simultaneous congruences

m s t — 1 (^rnod. iv\i ) , ( have a solution which is unique ) , i.e. a solution nn with 0 <■ m < n .

Then, since any multiple of an abundant number is also abundant, it follows that

ïY \ y r n — I ) ÏY1 — are N consecutive abundant numbers, each less than n . We therefore construct our sequence as follows. Let

fYI| — J 2 . . 3 ) rYl2_‘ 5 - " T . ^ ^ 2 , “ |) s ------J ^7. where m, is an abundant number, denotes the smallest prime such that , the product of primes from 5 to , i s an abundant number, is the prime following , and is the smallest prime so that , the product of primes between and j?*, i s an abundant number, and so on. It will follow that and are respectively the largest and smallest prime factors of nrir • (Clearly this construct ion is possible since by Lemma 3 diverges to -i-D4 , and therefore given any , it is possible to find a so that _ 1 I ( \ + X \ ^ 2 ), ^2r-l--- |>2J- fîij-

We now define

irJcrgn and observe firstly that, by Lemma 9, (3 .5 ) A < .. 2.“^" - n, and secondly that, by Lemma 7»

(3.6) - rr c^lorglogn^ where is a suitable absolute constant. Now we need only define tS by the pair of inequalities m, nrii ^ A ^ > and then, by ( 3 *5) i ^ m,nai- — ttih ïs r ^ n . Hence, (3«W issatisfied, and by the definition of me ,the numbers m,, mi,nr»M are relatively prime in pairs, so that there exists a sequenceof N consecutive integers, all ^-8.

abundant and le s s than n , It remains to show that N > C| icglcrg \ogn , for some absolute constant c, It is clear that A is a proper divisor of m , , since both are products of consecutive primes starting with 2, and so we can write

nn, nria^ % = A cl^

where cx> 1 , and A, cl) = 1 • Then, since all the m'5 are relatively prime in pairs we have, by (3*6), that

(3 .9 ) (T(m,) Crjra^ù. - (rfoQ > çXB) > , rvii *Y1 N+i A A

A lso, Since ^ ^ 1 ,

nn ù me /y>2.i

But as ^ is deficient, by definition of \=>2ii > the first factor on the right-hand side is less than 2. Also,

^ ’ fo r a l l b.- ,

and i t fo llo w s th at ( 3 . 8 ) < 3 , (^1= i,z, -

Hence, by (3»7) and (3.8), 3 *^*' > C^loglcrgn^

and we can therefore find an absolute constant c, such th at ( 3 .9 ) H > C, (oglogl< rgn , 4-9.

We have now proved that for sufficiently large n , there exist at least c, Icnglcrgicg o consecutive integers all abundant and le s s than o . We le t N be the len g th of the longest possible run of consecutive abundant numbers less than ^ n (i.e . let N be maximal), and we shall show that there e x is t s an absolute constant "> C/ , such th at

^3*10) M c^lorglorglogn. We proved our inequality (3*9) by showing, on one hand, that mi and on the o th er, that N j —f g( CgJog/ogn. 1=1 We shall prove (3.10) in a similar manner, i.e . by computing upper and lower bounds for the product

(3.11) T J ^ ) where, in this case m,, lYi,,... mr are a special sub-set of 1+ifc maximal set of N consecutive abundant numbers m, m -i,-... m-Ni-lj not exceeding n . (We note th at th is means th at ) . Firstly, since the m’s are abundant,

( 3 . 12) cr£mO 5 a.. me Secondly,

TT ( ' + -k- qmij)mi = TT (^ ' + pr ...... P

TT r )

r r ( '+ : 50.

We now let the numbers r)^be the integers between rv\ and m-N^i which are not divisible by any prime less than a certain fixed prime cy , to be determined later. Vfe denote by M the product

rv) I m2. rn I- , Then,

( 3 .1 3 ) Û ^ since at most [^] +• 1 of the integers noi are divisible by the prime . To compute an upper bound for the product on the right-hand side of ( 3 * 1 3 ) we divide the primes into two c la sse s. In the first we place those primes p satisfying

CJ, ^ H . For th ese, we have

[f] • ‘ f ' and it follows that rn_7 ^ I ^

hi M < TT

«"C t KM) ! • But

I Hh-o " ^ ' and therefore,

(3. 1M TT . e V, N 51.

In the second class we place all the primes ly , so that r f j ^ o -

For these primes we have, from Lemma 26 and Lemma

(3 .1 5 ) HM

< t t

t c, (Zlorgltjgn + loqH ^.

Hence, by ( 3 .1 3 ) , (3.1*t) and ( 3 .1 5 ),

TT otivu) ^ + (og N) ^ 1=1 me so that, from (3.12), ( 3 .16) ^ 2 f c ^ e (2!(yglcgn +

It is at this point that the reason for this particular method of proof becomes clean. Originally it may have seemed artificial to consider, instead of the entire run m-N+ 1 of consecutive abundant numbers, a particular sub-sequence nn,,ni2., nOr of integers not divisible (in a precise sense) by small primes, but in (3.16) we see the justification of this device. If the entire run were taken we should have N in place of r on the left of (3.16) 52. and 1 in place of on the right. In these circumstances the above lin e of argument breaks down, fo r ( 3 . I 6) i s then trivially true and yields no new information. On the other hand, by taking to be the least prime such that .

W8 have, by Lemma 273, that (3.17) r > 4^, providing M is sufficiently large (and since N > cdogfcgiogo we can choose n so large as to make this possible). By (3*17), we then have Z"- > e ^ ; so that, from (3.16),

& *= C 4 ^ /erg M J

< logltrgn . log H , since a+b -=-ab ,ifa>b> 2. • But for all sufficiently large n , 2-c^^ leg M < e . It then follows that

e ^ leg log n .

Hence, f in a lly , we can fin d a constant c.z.>osuch th at fo r a l l sufficiently large n ,

M < Cilogloglcrgr? ^ and th is com pletes the proof o f Theorem 11. 53.

3*3. In this section we consider the behaviour of the counting numbers of the sequences of abundant and primitive abundant numbers as functions of n • We shall denote by r DCn) r dé:ficic^nt' 1 A(n) the number oflabundant numbers not exceeding n , and by NCn) the number of primitive abundant numbers not exceeding n . More generally, we shall write A(n>A) for the number of X- abundant numbers not exceeding n • One of the most interesting questions to ask at this pèint is whether there is a fairly constant proportion of abundant numbers amongst the first n integers where n is large, i.e. does Irm e x is t? I f i t does, what proportion of these integers can we expect to be abundant numbers?

We can prove very simply the following theorem:

THEOREM 12. 6(n) - n 3 ^ i f n is sufficiently large.

Proof. Since, for an abundant number m we have

Therefore,

But in Theorem 7 we showed that

wi 5^.

and therefore, for sufficiently large n > f\M 6 tt^- - 1 + e n fe>

i . e . ^ f "

Since there are only 23 abundant numbers amongst the first 100 integers, it is very likely that much more than jxjjT foKn bevt |30ssiWfe. Theorem 12 is/tr^e. Certainly, if we consider solely the odd abundant numbers less than n , we can prove a result much better than would be expected from Theorem 12.

THEOREM 1 3 . R 'M < » n where denotes the number o f odd abundant numbers not exceeding n .

Proof. By an argument similar to that used in Theorem 12, we can show th at _L_ 2 ar^rYi-il ^ h'c^) _ L -h . ^ o • ro j n rns n JZm - 1 n n crdd where Tr(n) denotes the number of/^deficient numbers not exceeding n . But 2 . ? Msn ^"^'1 d |^ -i °CIP’ < 55.

. ^ 2. Since , and ^ ^ , we therefore have

(3.19) p- ^

Theorem I3 now follows from (3.18) and (3.I9), if n is sufficiently large.

Theorems 12 and 13 together suggest that there are fewer odd than even abundant numbers. This is certainly borne out by the limited numerical evidence at our disposal: there is only one odd abundant number in the f i r s t 1^000 in te g e r s, and eleven in the first 10,000, compared with 253 and 2601 even abundant numbers. However, this evidence suggests that the 8 ^ quotient lies approximately between O.23 and O.3. In 1933 Behrend 'proved, by an exceedingly difficult method, that in fact it lies between 0.2^1 and O.314. A year later Davenport Behrend and Chow3o, each proved independentlyy( the existence of the limit liM = {(X ), A > A, and further, that is a continuous function of X . Salie in a paper published much later, sharpened inequalities due to Behrend and proved th a t f > o-5fc9 , > o-2.^ 0-018 .

F. BEHREND. Berlin. Akad. Sitzungberichte (1933) 280-293* H. DAVENPORT. Ib id . (3-93^) 83O - 837.

H. SALIE. Math. Nachr. I k (1955) 39 - ^6. 56.

However, a year after Davenport had obtained his • ^ result, Erdos proved that ^ tends to a limit as n-a M , by a method which differs entirely from that of Davenport, and which requires only elementary considerations. He makes an ingenious use of the following lemma:

LEMMA 28.

Let ...... be an infinite sequence of positive integers, and let A,(n) denote the number of integers not exceeding 0 which are divisible by at least one , Then tends to a limit as n-? w providing only that converges.

Proof. Let A/ ’(n) denote the number of integers not exceeding n which are divisible by but not by any of the integers ivik., . Than

Since M = 3C + 0(1), it follows that

lirv, i V > ) ^ - L . - 2 . - L - + 5 .^

A. > s a y .

Now, trivially we have that

( 3 . 20) o » * Tt.rrifc ) h ^ ^6 nee (3.11) " ikc - \J. Lond. ifz P. ERDOS. Qnlonl Qtuarb. gkwtm. ^0. X1& .2-6 2 . 57.

It follows from ( 3 . 2 0 ) th a t i f 2 .-^ ^ converges,