{"@context":{"@language":"en","Affiliation":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","AggregatedSourceRepository":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","Campus":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","Creator":"http:\/\/purl.org\/dc\/terms\/creator","DateAvailable":"http:\/\/purl.org\/dc\/terms\/issued","DateIssued":"http:\/\/purl.org\/dc\/terms\/issued","Degree":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","DegreeGrantor":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","Description":"http:\/\/purl.org\/dc\/terms\/description","DigitalResourceOriginalRecord":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","FullText":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","Genre":"http:\/\/www.europeana.eu\/schemas\/edm\/hasType","GraduationDate":"http:\/\/vivoweb.org\/ontology\/core#dateIssued","IsShownAt":"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt","Language":"http:\/\/purl.org\/dc\/terms\/language","Program":"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline","Provider":"http:\/\/www.europeana.eu\/schemas\/edm\/provider","Publisher":"http:\/\/purl.org\/dc\/terms\/publisher","Rights":"http:\/\/purl.org\/dc\/terms\/rights","RightsURI":"https:\/\/open.library.ubc.ca\/terms#rightsURI","ScholarlyLevel":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","Title":"http:\/\/purl.org\/dc\/terms\/title","Type":"http:\/\/purl.org\/dc\/terms\/type","URI":"https:\/\/open.library.ubc.ca\/terms#identifierURI","SortDate":"http:\/\/purl.org\/dc\/terms\/date"},"Affiliation":[{"@value":"Science, Faculty of","@language":"en"},{"@value":"Mathematics, Department of","@language":"en"}],"AggregatedSourceRepository":[{"@value":"DSpace","@language":"en"}],"Campus":[{"@value":"UBCV","@language":"en"}],"Creator":[{"@value":"Kapoor, Vishaal","@language":"en"}],"DateAvailable":[{"@value":"2011-04-27T15:20:58Z","@language":"en"}],"DateIssued":[{"@value":"2011","@language":"en"}],"Degree":[{"@value":"Doctor of Philosophy - PhD","@language":"en"}],"DegreeGrantor":[{"@value":"University of British Columbia","@language":"en"}],"Description":[{"@value":"In this work we will consider several questions concerning the asymptotic\nnature of arithmetic functions. First, we conduct a finer analysis on the behavior\nof \u03bb(Euler's totient function(n)) in relation to \u03bb(\u03bb(n)), proving that log(\u03bb(Euler's totient function(n))\/\u03bb(\u03bb(n)))\nis asymptotic to (log log n)(log log log n)for almost all n. Second, we establish\nan asymptotic formula for sums of a generalized divisor function on the\nGaussian numbers. And third, for complex-valued multiplicative functions\nthat are suffciently close to 1 on the primes and bounded on prime powers,\nwe determine the average value over a short interval x < n \u2264 x+w provided\nthe interval is suffciently long with respect to x.","@language":"en"}],"DigitalResourceOriginalRecord":[{"@value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/34018?expand=metadata","@language":"en"}],"FullText":[{"@value":"Asymptotic Formulae for Arithmetic Functions by Vishaal Kapoor B.Sc., Simon Fraser University, 2004 M.Sc., The University of British Columbia, 2006 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2011 c\u00a9 Vishaal Kapoor 2011 Abstract In this work we will consider several questions concerning the asymptotic nature of arithmetic functions. First, we conduct a finer analysis on the be- havior of \u03bb(\u03d5(n)) in relation to \u03bb(\u03bb(n)), proving that log(\u03bb(\u03d5(n))\/\u03bb(\u03bb(n))) is asymptotic to (log log n)(log log log n) for almost all n. Second, we estab- lish an asymptotic formula for sums of a generalized divisor function on the Gaussian numbers. And third, for complex-valued multiplicative functions that are sufficiently close to 1 on the primes and bounded on prime powers, we determine the average value over a short interval x < n \u2264 x+w provided the interval is sufficiently long with respect to x. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Compositions of the Euler and Carmichael Functions . . . 8 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Notation and Useful Results . . . . . . . . . . . . . . . . . . . 12 2.3 The Proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . . 14 2.4 Large Primes q > Y . . . . . . . . . . . . . . . . . . . . . . . 16 2.5 Small Primes q \u2264 Y . . . . . . . . . . . . . . . . . . . . . . . 21 2.5.1 Normal Order of g(n) . . . . . . . . . . . . . . . . . . 24 2.5.2 Normal Order of h(n) . . . . . . . . . . . . . . . . . . 28 3 An Asymptotic Formula . . . . . . . . . . . . . . . . . . . . . . 30 4 Short Sums of Multiplicative Functions . . . . . . . . . . . . 37 iii 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.1 Compositions of the Euler and Carmichael Functions . . . . . 50 5.2 An Asymptotic Formula . . . . . . . . . . . . . . . . . . . . . 51 5.3 Short Sums of Multiplicative Functions . . . . . . . . . . . . 52 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 iv List of Figures 2.1 Ways that q can Divide \u03bb\u03bb(n) . . . . . . . . . . . . . . . . . . 17 2.2 Ways that q can Divide \u03bb\u03d5(n) . . . . . . . . . . . . . . . . . . 18 3.1 Keyhole Contour . . . . . . . . . . . . . . . . . . . . . . . . . 32 v Acknowledgements I am thankful for the brilliant teachers that I have had in my academic career. My supervisor, Greg Martin, has tirelessly taught me for the past 6 years, always being available to provide insight into mathematics and research. His encouragement and guidance have been most crucial to the completion of my MSc and Phd. My family has provided the solid support that has allowed me to pursue a higher education, and the continued love and laughter have made the pursuit all the more enjoyable (thanks Rosie). I would like to thank Erick Wong and Vasu Tewari for always providing interesting discussion about the curious questions of mathematics and com- puter trends, and Aurel Meyer for providing a great example to follow of solid work ethic. Lastly, I would like to thank Nike Vatsal, David Boyd, Michael Bennett, and Will Evans for providing valuable feedback and suggestions for improve- ment for this thesis; and I would like to thank Kevin Ford for suggesting the examples of Corollary 1. vi Dedication I dedicate this thesis to my family. vii Chapter 1 Introduction A central theme in multiplicative number theory involves determining the asymptotic behavior of functions of a multiplicative nature and of their sums over intervals of positive integers. The primary focus of this work is to derive asymptotic formulae for \u2022 the composition of Carmichael\u2019s \u03bb-function and Euler\u2019s totient func- tion, \u2022 a generalized divisor function on the Gaussian numbers, and \u2022 short interval sums of multiplicative functions of a certain type. A complex-valued function defined on the positive integers is called an arithmetic function. Such functions are very general, and typically one is interested in functions with more structure. If one has the additional rela- tionship f(ab) = f(a)f(b) when a and b are relatively prime positive integers, then f(n) is called a multiplicative function. We remind the reader that a prime number is a positive integer that has exactly two positive factors (i.e. 2, 3, 5, 7, 11, . . . ), and two positive integers are said to be relatively prime if they share no common prime factors. Let f(n) and g(n) be two arithmetic functions. We say that f(n) is asymptotically equivalent to g(n) (written f(n) \u223c g(n)) if f(n)\/g(n) tends to 1 as n tends to infinity; we may also express this equivalence as the asymptotic formula f(n) = (1 + o(1))g(n). Here we are using the little-oh notation f(n) = o(g(n)) which means that f(n)\/g(n) tends to 0 as n tends to infinity. Occasionally, one might obtain an asymptotic equivalence f(n) \u223c 1 g(n) on a subset of the natural numbers with asymptotic density 1. In such case we say that f(n) has normal order g(n). The study of arithmetic functions is central to number theory, and it is largely through the study of these functions that one gains insight into the positive integers. In particular, the multiplicative nature of the positive integers is of great interest, and studying arithmetic functions that have a multiplicative nature such as Euler\u2019s totient function and Carmichael\u2019s \u03bb- function can prove fruitful. Euler\u2019s totient function \u03d5(n) is defined as the cardinality of the multi- plicative group Z\/nZ, and Carmichael\u2019s function \u03bb(n) is defined to be the size of the largest cyclic subgroup of the multiplicative group Z\/nZ. We have \u03d5(n) = \u220f pa||n pa\u22121(p\u2212 1), where the product is over all prime powers pa exactly dividing n. Carmichael\u2019s \u03bb-function satisfies the relations \u03bb(n) = lcm(\u03bb(pa11 ), ..., \u03bb(p ak k )), where \u03bb(pa) = { pa\u22121(p\u2212 1) if p \u2265 3 or a \u2264 2, and 2a\u22122 if p = 2 and a \u2265 3. One notes from the definitions that \u03d5(n) \u2265 \u03bb(n), as the largest cyclic subgroup can be no larger than the group itself. Additionally, one sees that the primes that divide \u03bb(n) are exactly the same as those that divide \u03d5(n), except the primes divide \u03bb(n) with no larger multiplicity than they divide \u03d5(n). We remark that for odd n, the definitions of \u03d5(n) and \u03bb(n) are of a parallel nature. Namely, when n is odd, \u03d5(n) = \u220f pa||n p a\u22121(p \u2212 1) while \u03bb(n) = lcmpa||n{pa\u22121(p \u2212 1)}; for even n, the same relation holds up to a 2 factor of 2. It is helpful to keep such a description in mind when comparing \u03d5(n) and \u03bb(n). We have mentioned that by investigating \u03d5(n) and \u03bb(n) one gains insight into the multiplicative make-up of n. We give one such example that uses the following simple observation. When comparing the product of a set to the lowest common multiple of the same set, the larger the set, the more likely the lowest common multiple will be significantly smaller than the product due to the repeated occurrence of prime power factors. Thus if n is \u201chighly composite\u201d, that is, if n consists of the product of many distinct prime factors, then one would expect the ratio \u03d5(n)\/\u03bb(n) to be large \u2013 in this way we have a measure as to how \u201ccomposite\u201d n is. The ratio between \u03d5(n) and \u03bb(n) has been the study of [5, 8], and it has been shown that log(\u03d5(n)\/\u03bb(n)) has normal order (log log n)(log log log n). As Euler\u2019s totient function and Carmichael\u2019s \u03bb-function are of a funda- mental nature, we will consider a similar heuristic to analyse the multiplica- tive structure of the functions \u03d5(n) and \u03bb(n) themselves. We have mentioned that \u03d5(n) and \u03bb(n) are defined in a parallel fashion so one would expect a strong multiplicative similarity between these two functions. To argue such a case, we look at the comparison \u03bb(\u03d5(n))\/\u03bb(\u03bb(n)). Here we have applied \u03bb(n) to both \u03d5(n) and \u03bb(n) in the ratio \u03d5(n)\/\u03bb(n). What we find is that the logarithm log(\u03bb(\u03d5(n))\/\u03bb(\u03bb(n))) has the same normal order as that of log(\u03d5(n)\/\u03bb(n)). This is a curious result as we find that upon application of \u03bb(n), the normal order is unchanged. We prove that Theorem 1. The normal order of log(\u03bb(\u03d5(n))\/\u03bb(\u03bb(n))) is (log log n)(log log log n). We mention that the multiplicative nature of \u03d5(n) and \u03bb(n) has been pre- viously studied. In [7] it is shown that the number of prime factors (counted with multiplicity) of \u03d5(n) is normally the same as that of \u03bb(n) \u2013 both have normal order (log log n)2\/2. This is significantly larger than the number of prime factors of an arbitrary positive integer n which is of normal order log log n prime factors counted with multiplicity. By counting the primes 3 with multiplicity, one may deduce that the number of divisors of \u03d5(n) and \u03bb(n) has normal order exp((log 2)(log log n)2\/2) (see [11]). These results are suggestive of the multiplicative similarity of \u03d5(n) and \u03bb(n); however, the number of divisors of an integer, and the number of prime factors of an in- teger are functions which have smaller ranges for a given positive integer n than Carmichael\u2019s \u03bb-function. Thus the assertion of Theorem 1 provides more compelling evidence of the similarity between the multiplicative make- up of \u03d5(n) and \u03bb(n). Banks, Luca, Saidak, and Sta\u0306nica\u0306 [1] studied the set of n for which \u03bb(\u03d5(n)) = \u03d5(\u03bb(n)) as well as the asymptotic behavior of log(n\/\u03bb(\u03d5(n))). They proved that the normal order of log(n\/\u03bb(\u03d5(n))) is log(n\/\u03bb(\u03bb(n))), the latter having known normal order (log log n)2(log log log n) due to Martin and Pomerance [16]. Theorem 1 furthers the current state of knowledge con- cerning the composition of \u03bb(\u03d5(n)) by comparing \u03bb(\u03d5(n)) with the closer function \u03bb(\u03bb(n)) rather than the distant function n, and completes the pic- ture concerning the comparisons of the functions \u03d5(\u03d5(n)), \u03d5(\u03bb(n)), \u03bb(\u03d5(n)) and \u03bb(\u03bb(n)). Previously, the relative comparisons of each of these four func- tions were known with the exception of the comparison between \u03bb(\u03d5(n)) and \u03bb(\u03bb(n)) provided by Theorem 1. Our main innovation over the work of Banks, et al. in the analysis of \u03bb(\u03d5(n)) is the case-by-case analysis of the ways that the powers of primes can divide \u03bb(\u03d5(n)). Such an analysis requires many ingredients, the primary ingredients being the Tura\u0301n-Kubilius inequal- ity and a Brun-Titchmarsh inequality that gives upper bounds on the sum of 1\/p where p ranges through an arithmetic progression. The method we use is that of [5, 8, 16]. In addition to analysing the asymptotic behavior of an arithmetic func- tion, one may gain insight by analysing the asymptotic behavior of the sum of a function over intervals of positive integers. Such sums are revealing because a sum over many values of a function mitigates minor erratic behav- ior, typically resulting in well-behaved asymptotic formulae. One would be 4 hard-pressed to find a simple formula describing exactly which n are prime, but one can simply say there are asymptotically x\/ log x prime numbers less than x. Estimates for sums of arithmetic functions frequently arise in number theoretic applications so additional tools to handle these sums are valuable. As an example from analytic number theory, one commonly has to estimate sums of the generalized divisor function \u03c4k(n), that is, the number of ways of expressing a positive integer n as a product of k positive integers for some positive integer k. We now move to the second topic of this work. We will study a related divisor function which we call \u03c4 \u2032k(n) which counts the number of ways of expressing n as the product of k Gaussian numbers, that is, numbers which are the sum of two squares of integers. Our interest in this generalized divisor function has to do with an open problem concerning the distribution of Gaussian numbers in short intervals. It is currently known that for any positive real number x, the interval (x, x + 6x1\/4) contains a Gaussian number. It was our hope to gain more information about the indicator function for the Gaussian numbers, \u03c40(n), by studying the related functions \u03c4 \u2032k(n) for higher values of k in the same way that one may study a weighted sum to gain information about the original sum. What we have proven is in accordance with Landau\u2019s original work on the subject in 1908, where he states that the number of Gaussian numbers up to x is given by the asymptotic formula\u2211 n\u2264x \u03c4 \u20320(n) = \u03b1x(log x) \u22121\/2 +O(x(log x)\u22123\/2). Using the Selberg-Delange method, we establish an asymptotic formula for an even more general divisor function \u03c4 \u2032z(n) on the Gaussian numbers where z is allowed to be complex. We prove 5 Theorem 2. Let x \u2265 4. Then as x\u2192\u221e\u2211 n\u2264x \u03c4 \u2032z(n) = 1 \u0393(z\/2) x(log x)z\/2\u22121 +O(x(log x) x3\/5. Here \u03b5 > 0 is sufficiently small, and the implicit constant in the Vinogradov notation depends on A, \u03b8 and \u03b5. This theorem improves on the work of Bordelle\u0300s [2, 3] and is an attempt at furthering a general theory for short sums of multiplicative functions. An ideal result in this theory would be to prove that an asymptotic formula of the above type holds whenever the Euler product converged to a non-zero number. This would be analogous to the theory of long sums established by Delange and Hala\u0301sz (see [8]). Following a theorem of this sort, analogous re- sults for the generalized divisor function on the Gaussian numbers would be an important step towards understanding the distribution of Gaussian num- bers in short intervals. The author is currently investigating this direction of research. We will prove and provide detailed overviews of the presented theorems in the following three chapters. 7 Chapter 2 Compositions of the Euler and Carmichael Functions 2.1 Introduction Euler\u2019s totient function \u03d5(n) is defined to be the cardinality of the multiplica- tive group modulo n, for any positive integer n. Carmichael\u2019s \u03bb-function [4] denotes the cardinality of the largest cycle in the multiplicative group mod- ulo n. In other words, \u03bb(n) is the smallest positive integer m such that am \u2261 1 (mod n) for all reduced residues a (mod n). We notice that when the multiplicative group modulo n is cyclic, namely when n = 1, 2, 4, pa or 2pa where p is an odd prime and a \u2265 1, both \u03d5(n) and \u03bb(n) are equal. One may compute \u03d5(n) with the aid of the Chinese remainder theorem by using the formula \u03d5(n) = |(Z\/pa11 Z)\u00d7| \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 |(Z\/pakk Z)\u00d7| = pa1\u221211 (p1 \u2212 1) \u00b7 \u00b7 \u00b7 pak\u22121k (pk \u2212 1). where n has the prime decomposition n = pa11 \u00b7 \u00b7 \u00b7 pakk . For Carmichael\u2019s func- tion we note \u03bb(pa) = { pa\u22121(p\u2212 1) if p \u2265 3 or a \u2264 2, and 2a\u22122 if p = 2 and a \u2265 3, (2.1) together with \u03bb(n) = lcm(\u03bb(pa11 ), ..., \u03bb(p ak k )). (2.2) 8 We introduce notation that we will use in this chapter. Given two func- tions f(n) and g(n), we will frequently drop the outer parentheses from the expression f(g(n)), instead writing the composition as fg(n). Addi- tionally for f(n) denoting \u03bb(n), \u03d5(n) or log(n), we define f1(n) = f(n) and fk+1(n) = f(fk(n)) for k \u2265 1. We will use the expression \u201cfor almost all n\u201d to mean for n in a set of positive integers of asymptotic density 1. We will use the expression \u201cfor almost all n \u2264 x\u201d to mean that the number of counter- examples up to x is of size o(x). We recall that for arithmetic functions f(n) and g(n), we say f(n) has normal order g(n) if f(n) is asymptotic to g(n) for almost all n, or equivalently if f(n) = (1 + o(1))g(n) for almost all n. The theorem that we prove in this article is: Theorem 4. The normal order of log(\u03bb\u03d5(n)\/\u03bb\u03bb(n)) is log2 n log3 n. More precisely, we show that for almost all n \u2264 x, log \u03bb\u03d5(n) \u03bb\u03bb(n) = log2 n log3 n+O(\u03c8(x) log2 x), (2.3) where \u03c8(x) is a function tending to infinity slower than log3 x. We also show that the exceptional set of positive integers n for equation (2.3) is of asymptotic density O(x\/\u03c8(x)). There has been extensive study on the asymptotic behavior of \u03d5(n) and \u03bb(n) and their compositions. In 1928, Schoenberg [18] established that the quotient n\/\u03d5(n) has a continuous distribution function. In other words: Proposition 5. The limit \u03a6(t) = lim N\u2192\u221e |{n \u2264 N : n\/\u03d5(n) \u2265 t}|\/N exists and is continuous for any real t. Recently Weingartner [20] studied the asymptotic behavior of \u03a6(t) show- ing that as t tends to infinity, log \u03a6(t) = \u2212 exp(te\u2212\u03b3)(1 + O(t\u22122)), where \u03b3 = 0.5722... is Euler\u2019s constant. 9 We mention that higher iterates of \u03d5(n) have been studied by Erdo\u030bs, Granville, Pomerance and Spiro in [6]. They established: Proposition 6. The normal order of \u03d5k(n)\/\u03d5k+1(n) is ke \u03b3 log3 n, for k \u2265 1. In 1955 Erdo\u030bs established the normal order of log(n\/\u03bb(n)) in [5]. This re- sult was refined by Erdo\u030bs, Pomerance, and Schmutz in [8] where they proved the following result. Proposition 7. For almost all n, log n \u03bb(n) = log2 n(log3 n+ A+O((log3 n) \u22121+\u03b5), where A = \u22121 + \u2211 q prime q (q \u2212 1)2 = .2269688..., and \u03b5 > 0 is fixed but arbitrarily small. The author is currently undertaking the analysis of Theorem 1 to obtain a more accurate asymptotic formula of a form more closely resembling the previous proposition. Martin and Pomerance subsequently considered the question of under- standing the behavior of \u03bb\u03bb(n). In [16] they proved Proposition 8. For almost all n, log n \u03bb\u03bb(n) = (1 + o(1))(log2 n) 2 log3 n. (2.4) Based on heuristic reasoning, Martin and Pomerance have conjectured the behavior of the higher iterates of \u03bb\u03bb(n). Conjecture. For each k \u2265 1, log n \u03bbk(n) = ( 1 (k \u2212 1)! + o(1) ) (log2 n) k log3 n, 10 for almost all n. Banks, Luca, Saidak, and Sta\u0306nica\u0306 [1] studied the the compositions of \u03bb and \u03d5. In particular, they studied the set of n on which \u03bb\u03d5(n) = \u03d5\u03bb(n). In their paper, they also established the following: Proposition 9. For almost all n, log n \u03d5\u03bb(n) = (1 + o(1)) log2 n log3 n, and (2.5) log n \u03bb\u03d5(n) = (1 + o(1))(log2 n) 2 log3 n. (2.6) Consequently, log \u03d5\u03bb(n) \u03bb\u03d5(n) has normal order (log2 n) 2 log3 n. The proof of Proposition 9 uses a simple clever argument that rests on the theorem of Martin and Pomerance. It is interesting to see what we may obtain trivially from Propositions 8 and 9. Subtracting (2.5) from (2.4) gives an asymptotic formula for the comparison between \u03d5\u03bb(n) and \u03bb\u03bb(n), log \u03d5\u03bb(n) \u03bb\u03bb(n) \u223c (log2 n)2 log3 n, for almost all n. However, if we subtract (2.6) from (2.4), the main terms cancel and we are left with log \u03bb\u03d5(n) \u03bb\u03bb(n) = o((log2 n) 2 log3 n), for almost all n. This relation is interesting because it leads one to seek a more accurate asymptotic formula. This more accurate result is the content of Theorem 4. 11 2.2 Notation and Useful Results Let a, n \u2208 Z. Then the Brun-Titchmarsh inequality is the asymptotic rela- tionship that pi(z;n, a)\u001c z \u03d5(n) log(z\/n) (z > n), (2.7) where pi(z;n, a) is the number of primes congruent to a (mod n) up to z. We will be primarily concerned with implications of the Brun-Titchmarsh inequality in the case that a = 1. For convenience, define Pn to be the set of primes congruent to 1 (mod n), and for a given integer m, define the greatest common divisor of m and Pn, denoted (m,Pn), to be the product of the primes congruent to 1 (mod n) that divide m, or 1 if none exist. We will frequently use the following weaker form of (2.7) without mention. Lemma 10 (A Brun-Titchmarsh Inequality). For all z > ee, \u2211 p\u2264z p\u2208Pn 1 p \u001c log log z \u03d5(n) . (2.8) One may obtain (2.8) from (2.7) by partial summation. We will also use the following prime estimates stated in [16]. Lemma 11. Let z > e. Then we have the following: \u2211 p\u2264z log p\u001c z, \u2211 p\u2264z log p p \u001c log z, \u2211 p\u2264z log2 p p \u001c log2 z, \u2211 p>z log p p2 \u001c 1 z , and \u2211 p>z 1 p2 \u001c 1 z log z , These estimates follow via partial summation applied to Mertens\u2019 esti- mate M(z) = \u2211 p\u2264z(log p)\/p = log z + O(1). We illustrate the derivation of 12 the first tail estimate. One writes the Riemann-Steltjies integral \u2211 p>z (log p)\/p2 = \u222b \u221e z 1\/t dM(t) = M(t)\/t \u2223\u2223\u2223\u2223\u221e z + \u222b \u221e z M(t)\/t2 dt = (log z)\/z +O(1\/z) + \u222b \u221e z (log t)\/t2 +O(1\/t2) dt \u001c 1\/z2, as required. We remind the reader that we will be writing the composition of two arithmetic functions f(n) and g(n) as fg(n), and subscripts will be used with functions to indicate the number of times a function will be composed with itself (i.e. log2 n = log log n). The multiplicity to which a prime q divides n is denoted by \u03bdq(n). In what follows, the variables p, q, r will be reserved for primes. Throughout, we denote y = y(x) = log2 x. The function \u03c8(x) denotes a function tending to infinity, but slower than log y. When we use the expression \u201cfor almost all n \u2264 x\u201d, we will mean for all positive integers n \u2264 x except those in an exceptional set of asymptotic density O(x\/\u03c8(x)). We will make use of two parameters Y = Y (x) and Z = Z(x) in the course of the proof of Theorem 4 which we now define as Y = 3cy, and Z = y2, where c is the implicit constant appearing in the Brun-Titchmarsh theorem (2.7) and (2.8). 13 2.3 The Proof of Theorem 4 We intend to establish an asymptotic formula for log \u03bb\u03d5(n) \u03bb\u03bb(n) = \u2211 q (\u03bdq(\u03bb\u03d5(n))\u2212 \u03bdq(\u03bb\u03bb(n))) log q, (2.9) valid for n in a set of natural density 1. We will consider the \u201clarge\u201d q and the \u201csmall\u201d q separately. The cut-off for this distinction is the parameter Y giving the cases q > Y and q \u2264 Y , respectively. For q > Y , it will be unusual for \u03bdq(\u03bb\u03d5(n)) to be strictly larger than \u03bdq(\u03bb\u03bb(n)) and so the contribution in (2.9) from large q will be negligible. We bound the sum in (2.9) by the two cases,\u2211 q>Y (\u03bdq(\u03bb\u03d5(n))\u2212 \u03bdq(\u03bb\u03bb(n))) log q \u2264 \u2211 q>Y \u03bdq(\u03bb\u03d5(n))\u22652 \u03bdq(\u03bb\u03d5(n)) log q + \u2211 q>Y \u03bdq(\u03bb\u03d5(n))=1 (\u03bdq(\u03bb\u03d5(n))\u2212 \u03bdq(\u03bb\u03bb(n))) log q. (2.10) We prove the two bounds: Proposition 12. For almost all n \u2264 x,\u2211 q>Y \u03bdq(\u03bb\u03d5(n))=1 (\u03bdq(\u03bb\u03d5(n))\u2212 \u03bdq(\u03bb\u03bb(n))) log q \u001c y\u03c8(x), and Proposition 13. For almost all n \u2264 x,\u2211 q>Y \u03bdq(\u03bb\u03d5(n))\u22652 \u03bdq(\u03bb\u03d5(n)) log q \u001c y\u03c8(x). (2.11) 14 Combining Propositions 12 and 13 gives the upper bound we seek: Proposition 14. For almost all n \u2264 x,\u2211 q>Y (\u03bdq(\u03bb\u03d5(n))\u2212 \u03bdq(\u03bb\u03bb(n))) log q \u001c y\u03c8(x). (2.12) We now consider those primes q \u2264 Y. It will turn out that the main term comes from the quantity \u2211 q\u2264Y \u03bdq(\u03bb\u03d5(n)) with the sum \u2211 q\u2264Y \u03bdq(\u03bb\u03bb(n)) sufficiently small. Proposition 15. For almost all n \u2264 x,\u2211 q\u2264Y \u03bdq(\u03bb\u03bb(n)) log q \u001c y\u03c8(x). We are left with the final piece of establishing the asymptotic behavior of \u2211 q\u2264Y \u03bdq(\u03bb\u03d5(n)). This will involve a case-by-case analysis of the various ways that q can divide \u03bb\u03d5(n) with multiplicity. Two functions g(n) and h(n) arise from this analysis: g(n) = \u2211 q\u2264Y \u2211 \u03b1\u22651 q\u03b1+1|\u03d5(n) log q, h(n) = \u2211 q\u2264Y \u2211 \u03b1\u22651 \u03c9((n,Qq\u03b1 ))>0 log q, and Qq\u03b1 = {r \u2264 x : \u2203p \u2208 Pq\u03b1 st r \u2208 Pp}. We will show that g(n) is a good approximation to \u2211 q\u2264Y \u03bdq(\u03bb\u03d5(n)). To deal with g(n), we will choose a suitably close additive function to approximate g(n) and employ the Tura\u0301n-Kubilius inequality to find the normal order of g(n). 15 Proposition 16. For almost all n \u2264 x, g(n) = y log y +O(y). Proposition 17. For almost all n \u2264 x, h(n)\u001c \u03c8(x)y. We will combine these propositions to show Proposition 18.\u2211 q\u2264Y (\u03bdq(\u03bb\u03d5(n))\u2212 \u03bdq(\u03bb\u03bb(n))) log q = y log y +O(\u03c8(x)y). Summing the results from Propositions 14 and 18 gives\u2211 q (\u03bdq(\u03bb\u03d5(n))\u2212 \u03bdq(\u03bb\u03bb(n))) log q = y log y +O(\u03c8(x)y), which proves Theorem 1. In the following two sections, we will establish all of the propositions of this section except proposition 14 which we have established. 2.4 Large Primes q > Y In this section we prove Propositions 12 and 13. In order to proceed, we must first understand the different ways in which prime powers can divide \u03bb\u03bb(n) and \u03bb\u03d5(n). We assume Y \u2265 2 so all primes q under consideration are odd. From the definition \u03bb(n) (see (2.1) and (2.2)), one sees that \u03bb\u03bb(n) has q as a prime divisor if q2 divides \u03bb(n) or if n is divisible by some prime in Pq. We emphasize that these conditions are not exclusive. We may expand 16 these conditions in turn. If q2|\u03bb(n), then the higher power q3 divides n, or a prime in Pq2 divides n; while if some prime p \u2208 Pq divides \u03bb(n), then p2|n, or (n,Pp) > 1. We summarize these cases in the tree diagram below. q|\u03bb\u03bb(n) q2|\u03bb(n) \u2203p \u2208 Pq st p|\u03bb(n) q3|n \u2203p \u2208 Pq2 st p|n \u2203p \u2208 Pq st p2|n \u2203p \u2208 Pq st \u2203r \u2208 Pp, r|n Figure 2.1: Ways that q can Divide \u03bb\u03bb(n) We proceed with a similar analysis on the ways that q can be a divisor of \u03bb\u03d5(n). We saw that either q2 or some prime in Pq must divide the argument \u03d5(n) of \u03bb\u03d5(n). If two copies of q divide \u03d5(n), then their presence can come from the cube q3 dividing n, two distinct primes dividing n with each prime in Pq contributing one factor of q, both q2|n and a prime p \u2208 Pq dividing n, or a single prime in Pq2 dividing n. In the other case, if a prime p \u2208 Pq divides \u03d5(n), then p2|n or (n,Pp) > 1. Now we turn to the proof of Proposition 12. Proof of Proposition 12. One sees from the above analysis that q|\u03bb\u03d5(n) when- ever q|\u03bb\u03bb(n), so the only way (\u03bdq(\u03bb\u03d5(n)) \u2212 \u03bdq(\u03bb\u03bb(n))) can be nonzero is if q|\u03bb\u03d5(n) and q - \u03bb\u03bb(n). Moreover, there are only two ways that q can divide \u03bb\u03d5(n) but not \u03bb\u03bb(n); namely, two distinct primes p1, p2 \u2208 Pq could divide n, 17 q|\u03bb\u03d5(n) q2|\u03d5(n) \u2203p \u2208 Pq st p|\u03d5(n) q3|n \u2203p1, p2 \u2208 Pq st p1 6= p2, p1p2|n q2|n,\u2203p \u2208 Pq st p|n \u2203p \u2208 Pq2 st p|n \u2203p \u2208 Pq st p2|n \u2203p \u2208 Pq st \u2203r \u2208 Pp, r|n Figure 2.2: Ways that q can Divide \u03bb\u03d5(n) or both q2 and a single prime p \u2208 Pq could divide n. Thus 1 x \u2211 n\u2264x \u2211 q>Y \u03bdq(\u03bb\u03d5(n))=1 (\u03bdq(\u03bb\u03d5(n))\u2212 \u03bdq(\u03bb\u03bb(n))) log q \u2264 1 x \u2211 q>Y \u2211 p1,p2\u2208Pq p1p2|n n\u2264x log q + 1 x \u2211 q>Y \u2211 n\u2264x p\u2208Pq pq2|n log q \u001c 1 x \u2211 q>Y ( xy2 q2 + xy q3 ) log q \u001c y2\/Y, where we used Lemmata 10 and 11. Plugging in Y = 3cy the upper bound 18 is \u001c y. We deduce that for almost all n \u2264 x,\u2211 q>Y \u03bdq(\u03bb\u03d5(n))=1 (\u03bdq(\u03bb\u03d5(n))\u2212 \u03bdq(\u03bb\u03bb(n))) log q \u001c y\u03c8(x). Now we would like to show that\u2211 q>Y \u03bdq(\u03bb\u03d5(n))\u22652 \u03bdq(\u03bb\u03d5(n)) log q \u001c y2\u03c8(x)\/Y (2.13) holds normally. Proof of Proposition 13. Define Sq = Sq(x) = {n \u2264 x : q2|n or p|n for some p \u2208 Pq2} and S = \u222aq>Y Sq. A simple estimate shows that the cardinality of S is O(xy\/(Y log Y )). We will choose Y to be of asymptotic order \u001d y, thus the number of elements in S is O(x\/\u03c8(x)). As we are interested in a nor- mality result, we may safely ignore the positive integers in S. Consequently, to establish (2.13) for almost all n, it suffices to establish the mean value estimate 1 x \u2211 n\u2264x n6\u2208S \u2211 q>Y \u03bdq(\u03bb\u03d5(n))\u22652 \u03bdq(\u03bb\u03d5(n)) log q \u001c y2\/Y. (2.14) 19 To this end we write 1 x \u2211 n\u2264x n6\u2208S \u2211 q>Y \u03bdq(\u03bb\u03d5(n))\u22652 \u03bdq(\u03bb\u03d5(n)) log q \u2264 1 x \u2211 q>Y \u2211 n\u2264x n6\u2208S \u03bdq(\u03bb\u03d5(n))\u22652 \u03bdq(\u03bb\u03d5(n)) log q \u2264 1 x \u2211 q>Y \u2211 n\u2264x n 6\u2208S \u2211 \u03b1\u22652 q\u03b1|\u03bb\u03d5(n) 2 log q \u2264 2 x \u2211 q>Y \u03b1\u22652 \u2211 n\u2264x n6\u2208S q\u03b1|\u03bb\u03d5(n) log q \u2264 2 x \u2211 q>Y \u03b1\u22652 ( \u2211 n\u2264x p\u2208Pq\u03b1 p|\u03d5(n) + \u2211 n\u2264x n6\u2208S q\u03b1+1|\u03d5(n) ) log q. In order for the prime p to be a divisor of \u03d5(n), one of: p2 divides n, or r \u2208 Pp and r divides n for some prime r must occur. Thus, \u2211 n\u2264x p\u2208Pq\u03b1 p|\u03d5(n) 1 = \u2211 p\u2264x p\u2208Pq\u03b1 \u2211 n\u2264x p|\u03d5(n) 1\u001c \u2211 p\u2264x p\u2208Pq\u03b1 ( x p2 + \u2211 r\u2264x r\u2208Pp x r ) \u001c \u2211 p>q\u03b1 x p2 + \u2211 p\u2264x p\u2208Pq\u03b1 xy p \u001c x \u03b1q\u03b1 log q + xy2 q\u03b1 . (2.15) Summing over q > Y and \u03b1 \u2265 2 and weighting by log q we have the asymptotic upper bound 1 x \u2211 q>Y \u03b1\u22652 \u2211 n\u2264x p\u2208Pq\u03b1 p|\u03d5(n) log q \u001c y2\/Y. 20 Now we would like to establish 1 x \u2211 q>Y \u03b1\u22652 \u2211 n\u2264x n6\u2208S q\u03b1+1|\u03d5(n) log q \u001c y2\/Y. We note that the contribution of prime powers of q dividing \u03d5(n) for n 6\u2208 S can only come from distinct primes in Pq dividing n. We then have \u2211 n\u2264x n6\u2208S q\u03b1+1|\u03d5(n) 1\u001c 1 (\u03b1 + 1)! \u2211 p1,...,p\u03b1+1\u2208Pq \u2211 p1\u00b7\u00b7\u00b7p\u03b1+1|n\u2264x 1\u001c x(cy) \u03b1+1 (\u03b1 + 1)!q\u03b1+1 , (2.16) where we intentionally omit the condition that the primes pi \u2208 Pq are distinct and where c is the constant appearing in the Brun-Titchmarsh theorem. As Y \u2265 2cy we have cy\/q \u2264 1\/2. Thus summing the LHS of (2.16) over \u03b1 \u2265 2 and q > Y and weighting by log q gives \u2211 q>Y \u2211 \u03b1\u22653 xc\u03b1y\u03b1 \u03b1!q\u03b1 log q \u2264 xc2y2 \u2211 \u03b1\u22651 1 \u03b1!2\u03b1 \u2211 q>Y log q q2 \u001c xy2\/Y (2.17) as required. 2.5 Small Primes q \u2264 Y In this section we will be concerned with estimates for small primes; namely, we will prove Propositions 15, 16, 17 and 18. The main term in our asymp- totic formula will come from Proposition 16 which concerns the sum\u2211 q\u2264Y \u03bdq(\u03bb\u03d5(n)) log q. (2.18) The remaining two Propositions provide us with error terms. 21 We restate a Lemma 11 from [16] which we will use: Lemma 19. For a power of a prime qa, the number of positive integers n \u2264 x with qa dividing \u03bb\u03bb(n) is O(xy2\/qa). Proof of Proposition 15. We break the summation up into two parts depend- ing on the size of q\u03b1,\u2211 q\u2264Y \u03bdq(\u03bb\u03bb(n)) log q = \u2211 q\u2264Y log q \u2211 \u03b1\u22651 q\u03b1|\u03bb\u03bb(n) 1 \u001c \u2211 q\u2264Y log q \u2211 \u03b1\u22651 q\u03b1\u2264Z 1 + \u2211 q\u2264Y log q \u2211 \u03b1\u22651 q\u03b1>Z q\u03b1|\u03bb\u03bb(n) 1. We may bound the first sum as\u2211 q\u2264Y log q \u2211 \u03b1\u22651 q\u03b1\u2264Z 1\u001c Y logZ\/ log Y. We use an average estimate to bound the second sum. Note 1 x \u2211 n\u2264x \u2211 q\u2264Y log q \u2211 \u03b1\u22651 q\u03b1>Z q\u03b1|\u03bb\u03bb(n) 1 = 1 x \u2211 q\u2264Y log q \u2211 \u03b1\u22651 q\u03b1>Z \u2211 n\u2264x q\u03b1|\u03bb\u03bb(n) 1. (2.19) From Lemma 19, we see (2.19) is \u001c 1 x \u2211 q\u2264Y log q \u2211 \u03b1\u22651 q\u03b1>Z xy2 q\u03b1 \u001c \u2211 q\u2264Y y2 log q Z \u001c y 2Y Z . 22 Therefore \u2211 q\u2264Y log q \u2211 \u03b1\u22651 q\u03b1>Z q\u03b1|\u03bb\u03bb(n) 1\u001c y2Y \u03c8(x)\/Z, for almost all n \u2264 x. Combining our upper bounds gives\u2211 q\u2264Y \u03bdq(\u03bb\u03bb(n)) log q \u001c (Y logZ\/ log Y + y2Y\/Z)\u03c8(x), for almost all n \u2264 x. Substituting Y = 3cy and Z = y2 gives the theorem. Recall q\u03b1 divides \u03bb\u03d5(n) if one of \u2022 q\u03b1+1|\u03d5(n) \u2022 q\u03b1|p\u2212 1, p|r \u2212 1, r|n \u2022 q\u03b1|p\u2212 1, p2|n occurs. Note that these conditions are not mutually exclusive. We write (2.18) as \u2211 q\u2264Y \u03bdq(\u03bb\u03d5(n)) log q = g(n) +O ( h(n) + \u2211 q\u2264Y \u2211 p\u2208Pq\u03b1 p2|n log q ) , where g(n) = \u2211 q\u2264Y \u2211 \u03b1\u22651 q\u03b1+1|\u03d5(n) log q, h(n) = \u2211 q\u2264Y \u2211 \u03b1\u22651 \u03c9(n,Qq\u03b1 )>0 log q, and Qq\u03b1 = {r \u2264 x : \u2203p \u2208 Pq\u03b1 st r \u2208 Pp}. 23 Thus, for almost all n \u2264 x,\u2211 q\u2264Y \u03bdq(\u03bb\u03d5(n)) log q = g(n) +O(h(n) + \u03c8(x) log2 Y ). (2.20) In the next two sections, we prove Propositions 16 and 17. We see that Proposition 18 follows immediately by applying these two propositions to equation (2.20) giving\u2211 q\u2264Y \u03bdq(\u03bb\u03d5(n)) log q = y log y +O(y\u03c8(x)) for almost all n \u2264 x, as required. 2.5.1 Normal Order of g(n) Our strategy is to approximate g(n) from above and below by an additive arithmetic function, thus indirectly making g(n) amenable to the Tura\u0301n- Kubilius inequality. To start, write g(n) as g(n) = \u2211 q\u2264Y \u2211 \u03b1\u22651 q\u03b1+1|\u03d5(n) log q = \u2211 q\u2264Y (\u03bdq(\u03d5(n))\u2212 1) log q = \u2211 q\u2264Y \u2211 p|n \u03bdq(p\u2212 1) log q \u2212 Y (1 + o(1)) +O (\u2211 q\u2264Y \u03bdq(n) log q ) , (2.21) where we used the double inequality\u2211 p|n \u03bdq(p\u2212 1) \u2264 \u03bdq(\u03d5(n)) \u2264 \u2211 p|n \u03bdq(p\u2212 1) + \u03bdq(n). We will use the Tura\u0301n-Kubilius inequality: 24 Lemma 20 (The Tura\u0301n-Kubilius Inequality). There exists an absolute con- stant C such that for all additive functions f(n) and all x \u2265 1 the inequality\u2211 n\u2264x |f(n)\u2212 A(x)|2 \u2264 CxB(x)2 (2.22) holds where A(x) = \u2211 p\u2264x f(p)\/p, and B(x)2 = \u2211 pk\u2264x |f(pk)|2\/pk. Proof of Proposition 16. We will use Lemma 20 for the additive function g0(n) = \u2211 q\u2264Y \u2211 p|n \u03bdq(p \u2212 1) log q. Let A(x) and B(x) be the first and second moments: A(x) = \u2211 r\u2264x g0(r)\/r, and B(x) = \u2211 rk\u2264x g0(r k)2\/rk. Notice that g0(r k) = g0(r) = \u2211 q\u2264Y \u03bdq(r \u2212 1) log q leading to A(x) = \u2211 r\u2264x 1 r \u2211 q\u2264Y \u2211 p|r \u03bdq(p\u2212 1) log q = \u2211 q\u2264Y log q \u2211 r\u2264x \u03bdq(r \u2212 1) r = \u2211 q\u2264Y log q \u2211 \u03b1\u22651 \u2211 r\u2264x r\u2208Pq\u03b1 1 r . We split the sum over \u03b1 into \u2211 1\u2264\u03b1\u2264wq \u2211 r\u2264x r\u2208Pq\u03b1 1 r + \u2211 \u03b1>wq \u2211 r\u2264x r\u2208Pq\u03b1 1 r , 25 with wq to be determined later. The first we estimate with Page\u2019s theorem and the second we bound with the Brun-Titchmarsh bound\u2211 r\u2264x r\u22611 (mod d) 1\/r \u001c y\/\u03d5(d). \u221e\u2211 \u03b1=1 y \u03d5(q\u03b1) +O ( y qwq + wq ) = yq (q \u2212 1)2 +O ( y qwq + wq ) (2.23) Note that we used the bound 1\/qbwqc+1 = O(1\/qwq). Taking wq = log y\/ log q gives an error term of O(wq) = O(log y\/ log q). Summing (2.23) over q \u2264 Y weighted by log q gives the asymptotic formula A(x) = y \u2211 q\u2264Y q log q (q \u2212 1)2 +O ( Y log y log Y + Y ) = y log Y +O ( Y log y log Y + Y ) . (2.24) Expanding the square, write the second moment B(x) as B(x) = \u2211 q1,q2\u2264Y log q1 log q2 \u2211 r\u2264x \u03bdq1(r \u2212 1)\u03bdq2(r \u2212 1) \u2211 k\u22641 rk\u2264x 1\/rk. Uniformly in primes r, \u2211 k\u22651 1\/r k \u001c 1\/r. We may also express \u03bdqi(r \u2212 1) (i = 1, 2) as \u03bdqi(r \u2212 1) = \u2211 \u03b1i\u22651 r\u2208P q \u03b1i i 1, 26 giving the expanded B(x)\u001c \u2211 q1,q2\u2264Y log q1 log q2 \u2211 \u03b11,\u03b12\u22651 \u2211 r\u2264x r\u2208P q \u03b11 1 \u2229P q \u03b12 2 1 r . We split the sum in q1, q2 into the two cases: q1 = q2 and q1 6= q2. For the q1, q2 with q = q1 = q2 we have\u2211 q\u2264Y (log q)2 \u2211 \u03b11,\u03b12\u22651 \u2211 r\u2264x r\u2208P qmax(\u03b11,\u03b12) 1 r = \u2211 q\u2264Y (log q)2 \u2211 \u03b1\u22651 \u2211 r\u2264x r\u2208Pq\u03b1 \u03b1 r \u001c \u2211 q\u2264Y (log q)2 \u2211 \u03b1\u22651 \u03b1y q\u03b1 \u001c y \u2211 q\u2264Y (log q)2 q \u001c y(log Y )2. (2.25) If q1 and q2 are distinct then we have an upper bound (intentionally ignoring the condition that q1 6= q2 in the sum)\u2211 q1,q2\u2264Y log q1 log q2 \u2211 \u03b11,\u03b12\u22651 \u2211 r\u2264x r\u2208P q \u03b11 1 q \u03b12 2 1 r \u001c \u2211 q1,q2\u2264Y log q1 log q2 \u2211 \u03b11,\u03b12\u22651 y q\u03b111 q \u03b12 2 \u001c y \u2211 q1,q2\u2264Y log q1 log q2 q1q2 \u001c y(log Y )2. (2.26) Combining (2.25) and (2.26) gives B(x)\u001c y(log Y )2. (2.27) Using Lemma 20 we may conclude that The statement of Lemma 20 gives 27 us the equation \u2211 n\u2264x |g0(n)\u2212 A(x)|2 \u2264 CxB(x)2. (2.28) Thus the set of n \u2264 x on which g0(n) differs from A(x) by more than y is O(x(log Y )2\/y) = O(x\/\u03c8(x)). The mean value of \u2211 q\u2264Y \u03bdq(n) log q for n \u2264 x is\u001c 1\/x \u2211 q\u2264Y x log q\/q \u001c\u2211 q\u2264Y log q\/q \u223c log Y, so \u2211 q\u2264Y \u03bdq(n) log q \u001c log2 Y for almost all n \u2264 x. Thus from (2.21), we see that for almost all n \u2264 x, g(n) = y log Y +O ( Y log y log Y + Y ) , (2.29) Substituting Y = 3cy gives the theorem. 2.5.2 Normal Order of h(n) Proof of Proposition 17. In order to find an upper bound on a set of asymp- totic density 1, we will compute the first moment of h(n): H(x) := 1 x \u2211 n\u2264x h(n) = 1 x \u2211 q\u2264Y \u03b1\u22651 \u2211 n\u2264x \u03c9(n,Qq\u03b1 )>0 log q = 1 x \u2211 q\u03b1\u2264Z q\u2264Y \u03b1\u22651 \u2211 n\u2264x \u03c9(n,Qq\u03b1 )>0 log q + 1 x \u2211 q\u03b1>Z q\u2264Y \u2211 n\u2264x \u03c9(n,Qq\u03b1 )>0 \u03b1\u22651 log q. We deal with the two sums in turn. Small q\u03b1 The first part is for small powers of q: 1 x \u2211 q\u03b1\u2264Z q\u2264Y \u2211 n\u2264x \u03c9(n,Qq\u03b1 )>0 log q \u2264 1 x \u2211 q\u03b1\u2264Z q\u2264Y log q \u2211 n\u2264x 1 \u2264 \u2211 q\u03b1\u2264Z q\u2264Y log q = Y logZ log Y . (2.30) 28 Large q\u03b1 The second part is for large powers of q. In this case we use a crude estimate that is sufficient for our needs: 1 x \u2211 q\u03b1>Z q\u2264Y \u2211 n\u2264x \u03c9(n,Qq\u03b1 )>0 log q \u001c 1 x \u2211 q\u03b1>Z q\u2264Y log q \u2211 r\u2208Qq\u03b1 \u2211 n\u2264x r|n 1 \u001c 1 x \u2211 q\u03b1>Z q\u2264Y log q \u2211 r\u2208Qq\u03b1 x r \u001c \u2211 q\u03b1>Z q\u2264Y log q \u2211 p\u2208Pq\u03b1 \u2211 r\u2208Pp 1 r \u001c y2 \u2211 q\u03b1>Z q\u2264Y log q q\u03b1 . (2.31) The sum in the RHS of (2.31) is less than \u2211 q\u2264Y \u2211 \u03b1>logZ\/ log q log q\/q \u03b1 \u2264 2 \u2211 q\u2264Y log q\/q logZ\/ log q \u001c Y\/Z, or alternatively q\u03b1 \u2265 Z and\u2211q\u2264Y log q \u223c Y . Thus 1 x \u2211 q\u03b1>Z q\u2264Y \u2211 n\u2264x \u03c9(n,Qq\u03b1 )>0 log q = O(y2Y\/Z). (2.32) Summing (2.30) and (2.32) gives H(x)\u001c Y logZ\/ log Y + y2Y\/Z \u001c y, where we substituted the values of Y and Z. Thus, for almost all n \u2264 x, h(n)\u001c y\u03c8(x). 29 Chapter 3 An Asymptotic Formula for a Divisor Function on the Gaussian Numbers A natural number is a Gaussian number if it can be written as the sum of two squares of integers. We let S denote the set of Gaussian numbers and define S(x) to be the number of Gaussian numbers \u2264 x. In 1908, Landau proved that S(x) = \u03b1x(log x)\u22121\/2 +O(x(log x)\u22123\/2) where \u03b1 is the constant1 2\u22121\/2 \u220f p\u22613 (mod 4)(1\u2212 p\u22122)\u22121\/2. Under the Riemann hypotheses for \u03b6(s) and L(s, \u03c74) where \u03c74 is the non-principal character modulo 4, one can reduce the error term in Landau\u2019s problem to O(x1\/2(log x)2). Gediri [12] noted that though one can use the Riemann hypotheses to obtain a smaller power in the error term of Landau\u2019s problem, one can obtain an unconditional asymptotic formula for the summatory function of the number of Gaussian divisors with a small error term. Define the counting function for the number of Gaussian divisors, \u03c4 \u2032(n) = \u03a3\u2032n=ab1, and its relatives \u03c4 \u2032 k(n) = \u2211 s1s2...sk=n 1 for any positive integer k. Here both of the sums are over indices restricted to lie in S and \u03c4 \u2032(n) := \u03c4 \u20322(n). Gediri proves for positive integers k that D \u2032 2k(n) := \u2211 n\u2264x \u03c4 \u2032 2k(n) = xPk(log x) + O(x 1\u2212c\u2032\/k2\/3), where Pk(x) is a polynomial of degree k \u2212 1 and c\u2032 is an effective absolute constant. We will prove an analogous asymptotic formula for k any complex number, generalizing the definition of \u03c4 \u2032k(n) ap- propriately. First, we must define what we mean by \u03c4 \u2032k(n) for k \u2208 C. 1p is a prime unless otherwise specified. 30 Let F (s) = \u2211 n\u2208S n \u2212s be the Dirichlet function corresponding to the sequence of Gaussian numbers. Then for a complex number z \u2208 C the generalized Gaussian divisor function \u03c4 \u2032z(n) is defined by the relation F (s)z = \u221e\u2211 n=1 \u03c4 \u2032z(n) ns (~~ 1), where wz := exp(z logw). Unless otherwise specified, we will choose the logarithm to be real on the positive real axis. We prove the following theorem. Theorem 21. Let x \u2265 4. Then as x\u2192\u221e\u2211 n\u2264x \u03c4 \u2032z(n) = 1 \u0393(z\/2) x(log x)z\/2\u22121 +O(x(log x) 1. One sees that F (s)2 = \u03b6(s)L(s, \u03c74)\u03a6(s) where \u03c74 is the non- principal character modulo 4, and \u03a6(s) = (1\u22122\u2212s)\u22121\u220fp\u22613 (mod 4)(1\u2212p\u22122s)\u22121. 31 Therefore, F (s)z = ( 1\u2212 1 2s )\u2212z\/2 \u220f p\u22611 (mod 4) ( 1\u2212 1 ps )\u2212z\/2 \u220f p\u22613 (mod 4) ( 1\u2212 1 p2s )\u2212z\/2 = \u03b6(s)z\/2L(s, \u03c74) z\/2\u03a6(s)z\/2 = \u221e\u2211 n=1 \u03c4 \u2032z(n) ns . 0 1 iT -iT \u03a3 t Figure 3.1: Keyhole Contour 32 Recall the function log+ y defined to be log y for y \u2265 e and 1 otherwise. In this chapter, we will drop the superscript and simply refer to log y. Write s = \u03c3+it where \u03c3, t \u2208 R, define \u03c3(t) = 1\u2212c\/ log |t|, and take c > 0 sufficiently small so that \u03b6(s) and L(s, \u03c74) are non-zero in the region {s : \u03c3 > \u03c3(t)}. Define \u03ba = 1 + 1\/ log x, T = exp( \u221a log x). Perron\u2019s inversion formula [17, pp. 139-140] says that \u2211 n\u2264x \u03c4 \u2032z(n)\u2212 \u222b \u03ba+iT \u03ba\u2212iT F (s)z xs s ds\u001c \u2211 x\/2 0 one defines a Hankel contour H to be a contour formed from the path (\u2212\u221e,\u2212r] followed by the circle |s| = r (exclud- ing s = \u2212r) traced out counter-clockwise, followed by the path (\u2212\u221e,\u2212r], where we traverse (\u2212\u221e,\u2212r] twice, the first time with argument \u2212pi and 35 the second with argument pi. Though we do not use this fact, one has 1\/\u0393(w) = (2pii)\u22121 \u222b Hw \u2212zew dw for any complex w. For a proof see [19, pg 183]. If instead of (\u2212\u221e, r] we were to use the paths (\u2212X,\u2212r] traced out twice, we have what is called a truncated Hankel contour denoted by H(X). In other words, H(X) would consist of all the points on a Hankel contour with real part > \u2212X. We have the following result (see [19, pg. 184]). Lemma 22. Let X > 1. Then uniformly for w \u2208 C, 1 2pii \u222b H(X) w\u2212zew dw = 1 \u0393(z) +O(47|z|\u0393(1 + |z|)e\u2212X\/2). In the main term in (3.5) make the change of variables w = (s\u2212 1) log x. This transforms the contour integral into x(log x)z\/2\u22121 2pii \u222b H(c log x) w\u2212z\/2ew dw = 1 \u0393(z\/2) x(log x)z\/2\u22121 +O(x1\u2212c(log x)z\/2\u22121), (3.7) with implicit constant depending on z by the Lemma. Adding (3.7) to the error term (3.6) gives Theorem 21 since the remaining error terms are O(x\/(log x)R+2). 36 Chapter 4 Short Sums of Multiplicative Functions The literature is rich with asymptotic formulae for the sum of a multiplica- tive function over long intervals n \u2264 x. In contrast, little is known about multiplicative functions summed over short intervals x < n \u2264 x+y except in special circumstances. In 2002, Bordelle\u0300s showed that under the hypothesis that f(n) is a real-valued multiplicative function taking values in [0, 1] and assuming the value 1 on primes, f(n) has average value tending to Cf = \u220f p ( 1\u2212 1 p )( 1 + f(p) p + f(p2) p2 + ... ) on the short interval provided y \u2265 x1\/5+\u03b5 for some \u03b5 > 0. In this article, we extend Bordelle\u0300s theorem to a larger class of functions, lifting the requirement that f(n) be real-valued and weakening the hypothesis on the values of f(n) on the primes. More specifically, we allow any complex-valued multiplicative function that is uniformly bounded on the prime powers and \u201cclose\u201d to 1 on the primes. Theorem 23. Let A and \u03b8 be positive constants with \u03b8 \u2265 4\/5. Define FA,\u03b8 to be the class of complex-valued multiplicative functions f(n) which are bounded by A on prime powers and satisfy the inequality |f(p)\u2212 1| \u2264 Ap\u2212\u03b8, 37 for all primes p. For any f \u2208 FA,\u03b8 if \u03b8 \u2265 4\/5, then\u2211 x x3\/5. Here \u03b5 > 0 is sufficiently small, and the implicit constant in the Vinogradov notation depends on A, \u03b8 and \u03b5. When w \u2264 x3\/5, the three expressions x1\/5+\u03b5, x1\/15+\u03b5w2\/3, and x\u22121\/10+\u03b5w dominate the error term in the ranges w \u2264 x1\/5, x1\/5 < w \u2264 x1\/2, and w > x1\/2 respectively. In particular this theorem is nontrivial when w \u2265 x1\/5+\u03b5. We note that the range of \u03b8 can be enlarged to \u03b8 > 1\/2. However, in the range 1\/2 < \u03b8 < 4\/5 the error term depends heavily on the choice of \u03b8, so we omit the details and refer the reader to the comments following the proof of Theorem 23. For an application of Theorem 23, we find asymptotic formulae for short sums of the functions n\/\u03d5(n) and \u03c3(n)\/n over squarefree positive integers n. Both of these sums involve functions that are not identically 1 on the set of primes and that have values outside [0, 1] and thus can not be handled using Bordelle\u0300s theorem. Applying Theorem 23 with \u03b8 = 1 gives the short interval asymptotic formulae: Corollary 1. For 1 \u2264 w \u2264 x we have both\u2211 x 0. Both of these formulae are nontrivial when w > x1\/5+\u03b5. In what follows, we will not state the implicit dependence on A, \u03b8, or \u03b5 in the Vinogradov and big-Oh notations. We use the notation bac to denote the largest integer \u2264 a. The Dirichlet convolution of two arithmetic functions f and g is defined as (f \u2217 g)(n) = \u2211 ab=n f(a)g(b), where a, b range over positive integers. Recall a squarefree number is a natu- ral number not divisible by the square of any prime, and a squarefull number is a natural number with every prime factor appearing with multiplicity at least 2. We outline our strategy to prove Theorem 23. We begin by establish- ing two simple lemmata, and stating the more technical Lemma 26; this is followed by the proof of Theorem 23. We will require two results to prove Lemma 26 - we then state these results and prove the lemma. Lemma 24. Let \u03b5 > 0. For z \u2265 1 we have\u2211 m\u2264z m squarefull m\u03b5 \u001c z1\/2+\u03b5 (4.2) and we have \u2211 m>z m squarefull 1 m1\u2212\u03b5 \u001c 1 z1\/2\u2212\u03b5 (4.3) 39 if \u03b5 < 1\/2. Proof. We prove the second result - the proof of the first is analogous. Let S(t) be the number of squarefull numbers \u2264 t. Using integration by parts on the Riemann-Steltjies integral, \u2211 m>z m squarefull 1 m1\u2212\u03b5 = \u222b \u221e z 1 t1\u2212\u03b5 dS(t)\u001c S(t) t1\u2212\u03b5 \u2223\u2223\u2223\u2223\u221e z + \u222b \u221e z S(t) t2\u2212\u03b5 dt. (4.4) Golomb has shown that S(t) \u001c \u221at (see [13]), so (4.4) is \u001c z\u22121\/2+\u03b5 as required. Lemma 25. Suppose f \u2208 FA,\u03b8 for some A > 0, and \u03b8 \u2265 1\/2+\u03b5 where \u03b5 > 0. Let g = f \u2217 \u00b5. Then for y \u2265 1 we have\u2211 d\u2264y |g(d)| \u001c y1\/2+\u03b5 and we have \u2211 d>y g(d) d \u001c y\u03b5\u22121\/2 when \u03b5 < 1\/2. Proof. On primes p, |g(p)| \u2264 A\/p\u03b8 and on higher prime powers |g(pk)| \u2264 2A. Thus for squarefree n, one has g(n)\u001c n\u03b5\u2212\u03b8 (4.5) following from the well-known upper bound A\u03c9(n) \u001cA,\u03b5 n\u03b5. For arbitrary n, we have g(n)\u001c n\u03b5. (4.6) We use (4.5) and (4.6) to find upper bounds on the sums \u2211 d\u2264y |g(d)| and 40 \u2211 d>y g(d)\/d. One can uniquely parameterize d \u2264 y as the product of rela- tively prime m and n, with m squarefull, and n squarefree. Thus,\u2211 d\u2264y |g(d)| = \u2211 n\u2264y n squarefree |g(n)| \u2211 nm\u2264y (m,n)=1 m squarefull |g(m)| \u001c \u2211 n\u2264y n squarefree 1 n\u03b8\u2212\u03b5 \u2211 m\u2264y\/n m squarefull m\u03b5. By (4.2) this is \u001c \u2211 n\u2264y n squarefree 1 n\u03b8\u2212\u03b5 ( y n )1\/2+\u03b5 = y1\/2+\u03b5 \u2211 n 1 n\u03b8+1\/2 \u001c y1\/2+\u03b5 since \u03b8 > 1\/2. This establishes the first estimate. The deal with the second estimate, we write \u2211 d>y g(d) d = \u2211 n squarefree g(n) n \u2211 nm>y (m,n)=1 m squarefull g(m) m \u001c \u2211 n squarefree 1 n\u03b8+1\u2212\u03b5 \u2211 m\u2265y\/n m squarefull 1 m1\u2212\u03b5 . 41 By (4.3) this is \u001c \u2211 n squarefree 1 n\u03b8+1\u2212\u03b5 ( n y )1\/2\u2212\u03b5 \u001c y\u03b5\u22121\/2 \u2211 n 1 n\u03b8+1\/2 \u001c\u03b8 y\u03b5\u22121\/2, (4.7) when \u03b5 < 1\/2. The inner sum converges because \u03b8 > 1\/2. Lemma 26. There exists a sufficiently small positive constant c0 such that if Y \u2264 c0X1\/2 and X \u2265 1\/2, then \u2211 Y 0. Lemma 26 is based on lattice point estimates of Huxley and Sargos (see [14, 15]), and Filaseta and Trifonov [10, Theorem 7]; we summarize these results in Lemma 28. Lemma 26 is essentially proved by Bordelle\u0300s over the two papers [2, 3]. For the convenience of the reader, we will collect the details of the proof following the proof of Theorem 23. Now, we are ready to prove Theorem 23. Like Bordelle\u0300s, we express f(n) as the Dirichlet convolution of the identity function and a second function, g(n), which is small in magnitude in lieu of the hypotheses imposed on f(n). However, because we have relaxed the conditions on f(n) at primes, we are left with more technical estimates on g(n). Further, we will consider short intervals of length \u2264 x, where Bordelle\u0300s considers intervals of length \u001c x1\/2. In this wider consideration, we will use the estimate of Filaseta and Trifonov (4.18) in as large a range as possible, and use the estimates of Huxley and Sargos (4.17) otherwise. 42 Proof of Theorem 23. Let g = f \u2217 \u00b5 and 1 \u2264 y \u2264 x. Write\u2211 xy g(d)\/d \u2223\u2223\u2223\u2223)+O(\u2211 d\u2264y |g(d)| ) . (4.9) Applying Lemma 25 to the error terms and expressing the infinite sum as an Euler product gives \u03a31 = yCf +O(y1\/2+\u03b5), (4.10) where Cf = \u220f p(1\u2212 1\/p)(1 + f(p)\/p+ f(p2)\/p2 + ...). Now we consider \u03a32: |\u03a32| \u2264 \u2211 n squarefree n\u22642x |g(n)| \u2211 y\/nu g(k) k ) . Applying the estimates of Lemma 25 gives \u2211 n\u2264u f(n) = u \u221e\u2211 k=1 g(k) k +O(u1\/2+\u03b5) = uCf +O(u 1\/2+\u03b5). Applying this result for u = x+ w and u = x and subtracting gives\u2211 x\u2264n\u2264x+w f(n) = wCf +O(x 1\/2+\u03b5), 45 as required. In Theorem 23, although one requires \u03b8 \u2265 4\/5 for the bound on \u03a32 in Equation (4.11), we remark that some mileage may be obtained in the range 1\/2 < \u03b8 < 4\/5. In this range of \u03b8, one may sum the series involving n up to x to obtain the upper bound \u03a32 \u001c x3\u03b5E \u2032\u03b8(x, y) + x2\u03b5E(x, y), (4.14) where E \u2032\u03b8(X, Y ) = X1\/3\u2212\u03b8Y 2\/3 +X11\/15\u2212\u03b8Y 4\/15 +X1\u2212\u03b8 +X5\/6\u2212\u03b8Y 1\/6. Here we have used the bound \u2211 n\u2264u 1\/n a \u001c 1 + u1+\u03b5\u2212a valid for a > 0 and including a u\u03b5 to bound log u in the case a = 1. One may then proceed similarly as in the proof of Theorem 23, again using standard convolution arguments applied to the long sums when appropriate. We omit the details as they depend on the choice of \u03b8. The proof of Lemma 26 requires the following: Lemma 27. For 1 \u2264 y \u2264 x, and r a positive integer, one has T (x, y; r) := \u2211 x 0. We 46 define R(\u03d5,N, \u03b4) = |{n \u2208 Z : N < n \u2264 2N and \u2203m \u2208 Z st |\u03d5(n)\u2212m| \u2264 \u03b4}|. Then for a function \u03b3(n) > 0 with \u03b4 = \u03b4(N) = maxN 0. If there exists \u03bbk = \u03bbk(N) such that |\u03d5(k)(x)| \u0010 \u03bbk when N < x \u2264 2N , then R(\u03d5,N, \u03b4)\u001c N\u03bb2\/k(k+1)k +N\u03b42\/k(k\u22121) + (\u03b4\u03bb\u22121k )1\/k + 1. (4.17) There exists a sufficiently small positive constant c0 = c0(k) such that if Nk\u22121\u03b4 \u2264 c0 and N \u2264 x1\/k, then R(x\/nk, N, \u03b4)\u001ck x1\/(2k+1) + x1\/(6k+3)\u03b4N (6k2+k\u22121)\/(6k+3). (4.18) Proof of Lemma 26. We sketch the proof from \u201cOn Short Sums of Certain Multiplicative Functions\u201d and \u201cCorrigendum to On Short Sums of Certain Multiplicative Functions.\u201d The sum over squarefull m in (4.8) can be parameterized in terms of a, b writing m = a2b3 with b squarefree. Since m > Y , one of a and b is greater 47 than Y 1\/5. Thus\u2211 Y ~~