AN ESSAY ON THE DISTRIBUTION OF VALUES OF THE CT-FUNCTION ' AND RELATED TOPICS ANITA STRAKER ProQuest Number: 10096394 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest. ProQuest 10096394 Published by ProQuest LLC(2016). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. Microform Edition © ProQuest LLC. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 PREFACE 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 extensions which 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 <x and b • Thus (a,6) = 1 means that cx and b are co-prime. denotes the binomial coefficient (n-oi r t '5(s), the Riemann-zeta function, denotes "As , where > 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 <c Hence, 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><XN+l pf=N+l n-.(-Tsn+i C ^ N+"» ^ M - 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.