Abstract Analytic Number Theory

By John Knopfmacher

"This book is well-written and the bibliography excellent," declared Mathematical Reviews of John Knopfmacher's innovative study. The three-part treatment applies classical analytic number theory to a wide variety of mathematical subjects not usually treated in an arithmetical way. The first part deals with arithmetical semigroups and algebraic enumeration problems; Part Two addresses arithmetical semigroups with analytical properties of classical type; and the final part explores analytical properties of other arithmetical systems.
Because of its careful treatment of fundamental concepts and theorems, this text is accessible to readers with a moderate mathematical background, i.e., three years of university-level mathematics. An extensive bibliography is provided, and each chapter includes a selection of references to relevant research papers or books. The book concludes with an appendix that offers several unsolved questions, with interesting proposals for further development.

• Language: English
• Publisher: Dover Publications
• Release date: Mar 17, 2015
• ISBN: 9780486169347

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THEORY

INTRODUCTION

The purpose of this introduction is to give a rapid survey, without much specific detail, of the aims of this book and the kinds of topics that it covers.

Abstract analytic number theory seems to have arisen first as a generalization of the classical number theory of the positive rational integers, with special emphasis on the derivation of generalizations of the famous

Prime Number Theorem. π(x)~x/log x as x→∞, where π(x) denotes the total number of positive rational primes p x.

Such investigations may have been motivated partly by the search for an elementary proof of the Prime Number Theorem, which was not found until the latter part of the decade 1940—1950, and perhaps partly by the fact that they could be used to simultaneously cover at least part of Landau’s classical Prime Ideal Theorem, which extends the Prime Number Theorem to ideals in an algebraic number field. In this way, with ordinary rational integers or ideals in algebraic number fields as essentially the only significant examples, and with the major interest centering around the stability of classical theorems (i.e. the extent to which they remain true under weakened abstract hypotheses), one might perhaps view the abstract theory as largely an internally motivated variation of part of classical analytic number theory.

Recently, however, it has been noted that the ideas and techniques of the above and other parts of classical analytic number theory have a bearing on and applications to a wide variety of mathematical systems quite distinct from the multiplicative semigroups formed by the positive integers or the ideals in a given number field K.

In the first place, it had been apparent for many decades that there are a large number of classes of mathematical systems, particularly ones arising in abstract algebra, which have elementary unique factorization properties analogous to those of the positive integers 1, 2, 3, . . . . For example, this is basically the content of theorems of the Krull–Schmidt type, which are known to be valid for many categories of mathematical objects. However, until very recently, the analogy mentioned was usually taken to both begin and end with the trivial algebraic concept of unique factorization.

of all finite abelian groups.

not only the simple unique factorization properties implied by the Fundamental Theorem on Finite Abelian Groups but also many asymptotic conclusions of the analytic theory of numbers. Apart from any interest provided by the existence of such analogies in a specific and natural non-classical arithmetical system, such results also yield asymptotic enumeration theorems regarding the average numbers of isomorphism classes of finite abelian groups with certain natural types of properties. Theorems of the latter kind could also be considered to be of intrinsic interest, regardless of any abstract arithmetical interpretation. In fact, Cohen’s work depends heavily on an earlier enumeration theorem of Erdos and Szekeres, which provides an asymptotic mean-value for the total number a(n) of all isomorphism classes of groups of order n as nthis provides an asymptotic enumeration result concerning the indecomposable finite abelian groups.)

is only the first of infinitely many non-classical but perfectly natural arithmetical systems, which may be investigated fruitfully with the aid of methods of ordinary analytic number theory. Surprisingly perhaps, it turned out that in many cases the previously-mentioned abstract theory of numbers provides a most appropriate tool, which not only brings out the analogies of the new systems with ordinary multiplicative number theory but can also be used to derive some deep asymptotic enumeration theorems as well — principally as corollaries of an Abstract Prime Number Theorem subject to certain hypotheses. Further development of the previous abstract theory then also leads to other enumeration theorems in various special cases of interest. In this connection, however, it should be emphasized that these enumeration theorems only become corollaries of the abstract discussion after certain (usually non-trivial) theorems have been proved, which show that the appropriate abstract hypotheses are actually valid for the particular systems considered; a similar comment applies to Landau’s Prime Ideal Theorem mentioned earlier.

Most of the non-classical types of arithmetical systems referred to here arise by considering the isomorphism classes of objects in certain particular categories of interesting mathematical objects. In a variety of cases, however, the asymptotic behaviour of the number of isomorphism classes of objects of a certain type and size is different from that suggested by analogy with ordinary number theory. By a remarkable coincidence, many systems of this kind turn out to be amenable to the techniques and results of a quite distinct branch of classical number theory — the branch known as additive analytic number theory. Thus, although the earlier abstract theory is then no longer applicable, another field of number theory — which was initiated for very different purposes by Hardy and Ramanujan around 1917 — turns out to provide tools of precisely the type now required. Some of the resulting arithmetical conclusions contrast strongly with the corresponding facts valid for the positive integers and their closer analogues; in particular there now arises an Additive Abstract Prime Number Theorem involving asymptotic statements of a different kind from those of the previous theorems in this direction.

The purpose of this book is to provide a detailed introduction to the topics outlined above. In this respect, it does not attempt to give an exhaustive account of all or the sharpest known results in this area in every case. However, by dealing carefully with some of the more fundamental concepts and theorems, it is hoped that the treatment will both convince the reader of the reasonably wide scope of the ideas discussed and of their interest for the particular systems considered in detail, and be accessible to readers with only a moderate mathematical background. Roughly three years’ experience of university mathematics should be perfectly ample for most purposes below, although in some cases the text will refer to concepts and theorems involving topics (like those of algebraic number theory or the theory of Lie groups and symmetric Riemannian manifolds, for example) which may not be familiar to a reader with only this background. However, such references to topics of a more advanced kind occur mainly in the description of natural examples that provide motivation for the later abstract discussion. The treatment of such concrete examples (which makes up most of Chapter 1) is intended to stress the rather common occurrence of arithmetical systems in many branches of mathematics — a fact which, in this book, is taken as basic motivation for the abstract theory. Nevertheless, it is not necessary for immediate purposes that the reader be thoroughly familiar with all such specific systems. (Actually, the outline descriptions of parts of relatively advanced subjects that are included at certain stages in this book should usually suffice for immediate needs. In the case of the examples of Chapter 1 especially, we suggest that to begin with the reader should merely glance fairly rapidly at the descriptions given, then refer to these again when references are made in later chapters, and only consider detailed treatments of such background topics when this is convenient or a specific need arises.)

There is one particular field with which a moderate familiarity on the part of the reader would perhaps be useful for present purposes. This is classical multiplicative analytic number theory — a subject which can easily be introduced in its initial stages during the early years of university mathematics, although it is not always included. (The levels in analytic number theory reached in the books of Hardy and Wright [1] and Chandrasekharan [1], for example, should be more than adequate for a reader of the present text.) Even here, although some previous knowledge in this direction would probably be helpful, it is worth remarking that in principle such familiarity is not strictly essential. For, in the first place, it will be trivially obvious that certain abstract hypotheses considered below are valid for the ordinary positive integers, and, secondly, that part of the abstract theory which covers the rational integers as a special case is often only slightly more difficult at the level of useful generality than the restricted theory for the classical case alone. Thus, in theory, the present treatment could also be used as an introduction to classical analytic number theory, although it is not suggested that it should be so used — except perhaps by readers who are already fairly sophisticated in other branches of mathematics. (Even such readers might find it illuminating to refer on occasion to a more classical type of text, such as one of those mentioned above, say.)

The plan of the book is as follows: The general arithmetical systems that have been referred to give rise in the first place to the concept of an arithmetical semigroup. This is defined in Chapter 1. After a discussion of various natural examples of arithmetical semigroups, the rest of Part I is concerned mainly with algebraic properties of such semigroups and complex-valued arithmetical functions on them, and with the application of some of the conclusions to questions of enumeration. In particular, Part I deals with the algebraic enumeration of the isomorphism classes of objects in various specific mathematical categories of interest. Here, the term algebraic enumeration is used to indicate that the numbers involved are shown to be in principle computable from identities for the coefficients of certain explicit formal generating functions or series. Questions of asymptotic enumeration of the kinds mentioned earlier often become more amenable when such algebraic relations are already known, but in themselves the results of algebraic enumeration usually provide no insight into the asymptotic conclusions that may be valid. It is for this purpose that methods of analytic number theory often become relevant — even for a mathematician who might wish to solve a particular problem of asymptotic enumeration without having a special interest in number theory per se. Such asymptotic questions are included amomgst the topics treated in the second and third parts of the book, on the basis of some of the algebraic results of Part I.

Part II of the book deals with arithmetical semigroups having analytical properties of a kind similar to those of the classical semigroups of all positive integers or non-zero integral ideals in a given algebraic number field K. A portion of the discussion is concerned with establishing the validity of a certain Axiom A for the particular semigroups associated with various interesting categories of objects. This in itself amounts to the proof of a number of asymptotic enumeration theorems regarding the average numbers of isomorphism classes of objects of large cardinal in those categories. Apart from this, the relevant chapters investigate consequences of the abstract Axiom A, including the first Abstract Prime Number Theorem discussed above.

Part III contains two chapters, of which the first develops the analytical theory of additive arithmetical semigroups of a certain type. These semigroups are the ones implicitly referred to above in connection with the classical work of Hardy and Ramanujan in the field of additive analytic number theory. The final chapter treats arithmetical formations, which are systems whose theory allows both a generalization and a refinement of the theory of abstract arithmetical semigroups. Put roughly, an arithmetical formation consists of an arithmetical semigroup together with an equivalence relation on the semigroup which partitions it into classes that in many ways are analogous to arithmetical progressions of positive integers. The two chapters of Part III each contain proofs of further abstract prime number theorems, of kinds appropriate to the types of system discussed in either case.* Once again, asymptotic enumeration theorems play roles which are often of intrinsic interest regardless of the abstract number-theoretical background. In addition, some of these theorems are useful in establishing the validity of relevant abstract axioms for special systems of interest, while others occur as corollaries of the abstract discussion itself (after the axioms have been verified).

Most of the material below has not appeared previously in book form, and some of it is at most only implicit in the relevant research articles covering the field. In writing the text, it has seemed advisable not to attempt to give credit or assign priority to the particular authors whose work may be related to given theorems, except in a few special cases (usually ones in which the named authors had proved the particular results for the ordinary integers). A procedure of this kind is probably always open to debate, and it is in no way implied that the names of relevant authors do not deserve to be attached to theorems in other instances. (In certain cases, such as that of the Abstract Prime Number Theorem based on Axiom A, a special difficulty arises: the number of relevant authors is very large and difficult to evaluate. For example, see the lengthy history of the classical Prime Number Theorem described in Landau’s Handbuch [10].)

In order to at least partly counter-balance the present approach to the history of the subject, a fairly extensive bibliography is provided, and at the end of each chapter there is a selection of references to research papers or books with some bearing on, or interest related to, the contents of the chapter. It is worth remarking that the bibliographical references which occur in the text are usually intended to point the way to developments or topics of related interest, that are not covered fully in this book; their presence at those places is not intended to convey any deeper significance. In addition to containing directly relevant material, the bibliography also includes a selection of further references to topics of related interest which are not mentioned in the text: e.g. the zeta functions of rational simple algebras, of algebraic varieties, and of diffeomorphisms of manifolds.

For the expert on analytic number theory in particular, it may be emphasized again that the present book is concerned mainly with outlining the general scope of certain ideas and methods of analytic number theory within a variety of different mathematical fields besides classical number theory; there is no attempt at deriving the best possible or sharpest known asymptotic estimates in every case. (For information about sharper estimates in the analytic theory of the positive integers, we refer in particular to the quite recent books of Chandrasekharan [2], Huxley [1], Prachar [2] and Schwarz [4]; for information about the further analytic theory of algebraic number fields, particular reference may be made to the books of Goldstein [1], Landau [9], Lang [2] and Narkiewicz [1].)

A question that an expert might perhaps ask concerns whether the present development of abstract analytic number theory is sufficiently ripe for the publication of an account in book form. In this connection, the author believes that the present development of the subject is certainly ripe enough for the writing of an introduction to the field, and this is what this book attempts to provide. Examination of the text and articles listed in the bibliography will show that there are many aspects of this field that remain open to further investigation, and hopefully the subject will continue to develop rapidly in the near future. (A selection of unsolved questions is included at the end of the book.)

Comment on notation. A reference in the text of the type Proposition 4.3.7 indicates Proposition 3.7 of Chapter 4, § 3; a reference of the type Proposition 2.6 indicates Proposition 2.6 of § 2 of the chapter in which the reference occurs.

---

* A quite extensive class of natural arithmetical semigroups with still different asymptotic properties is treated by the author [16].

PART I

ARITHMETICAL SEMIGROUPS AND ALGEBRAIC ENUMERATION PROBLEMS

Part I is concerned with the algebraic properties of arithmetical semigroups and arithmetical functions on such semigroups, and with the application of conclusions about these properties to questions of enumeration. In particular, it deals with the enumeration of the isomorphism classes of objects in various specific categories of interest. This type of enumeration is of algebraic nature, involving the coefficients of formal generating functions or series of certain kinds. Questions of asymptotic enumeration, dealing particularly with the asymptotic behaviours of the numbers of isomorphism classes of objects of large cardinal or dimension in various natural categories, are included amongst the topics investigated in Parts II and III on the foundation of the algebraic results of Part I.

CHAPTER 1

ARITHMETICAL SEMIGROUPS

This chapter discusses a selection of natural examples of concrete mathematical systems that have arithmetical properties which are closely akin to those of the positive integers 1, 2, 3, ... in certain ways. The examples given split into two types. Firstly, there are algebraic systems arising from the study of rings that include or generalize the semigroup of positive integers itself. Secondly, we consider a selection of examples which arise from specific categories of mathematical objects for which a theorem of the Krull–Schmidt type is valid. It is the extension of methods of classical analytic number theory, so as to encompass such examples associated with explicit categories, which provides the abstract treatment of the later chapters with its most convincing justification, perhaps.

§ 1. Integral domains and arithmetical semigroups

We begin with a formal definition of the elementary but fundamental concept to be studied below.

Let G denote a commutative semigroup with identity element 1, relative to a multiplication operation denoted by juxtaposition. Suppose that G has a finite or countably infinite subset P (whose elements are called the primes of G) such that every element a ≠ 1 in G has a unique factorization of the form

where the pi are distinct elements of P, the αi are positive integers, r may be arbitrary, and uniqueness is understood to be only up to the order of the factors indicated. Such a semigroup G will be called an arithmetical semigroup if in addition there exists a real-valued norm mapping | | on G such that:

(i) |1| = 1, |p| > 1 for p P,

(ii) |ab| = |a||b| for all a,b G,

(iii) the total number NG(x) of elements a G of norm |ax is finite, for each real x > 0.

It is not difficult to verify that the conditions (i)-(iii) are equivalent to conditions (i) and (ii) together with

(iii)′ the total number πG(x) of elements p P of norm |px is finite, for each real x>0.

Before turning to the discussion of natural examples of arithmetical semigroups, it may be emphasized that it is especially the finiteness conditions (iii), (iii)′ that are crucial in our use of the adjective ‘arithmetical’, and in the later consideration of methods of analytic number theory in order to obtain information about special semigroups of interest.

1.1. Example. The pro to-type of all arithmetical semigroups is of course the multiplicative semigroup Gz of all positive integers {1,2, 3,...}, with its subset Pz of all rational primes {2, 3, 5, 7,...}. Here we may define the norm of an integer n to be |n|=n, the greatest integer not exceeding x.

Although the function Nz(xremains mysterious to this day. The asymptotic behaviour of π(x) for large x forms the content of the Prime Number Theorem, which states that

A suitably generalized form of this theorem will be proved in Part II.

1.2. Example: Euclidean domains. A simple way in which unique factorization arises in elementary abstract algebra is by means of the study of Euclidean domains, or more generally principal ideal domains.

If D denotes an integral domain, two elements a,b D are called associated if and only if a=bu for some unit u D. The relation of being associated is an equivalence relation on D, and it is easy to see that the resulting set GD of all associate classes ā of non-zero elements a D In the case when D is a principal ideal domain, the content of the Unique Factorization Theorem for D of GD of prime elements p D.

If, in addition, D is a Euclidean domain with norm function | |, one may define a norm on GD satisfying conditions (i) and (ii) above by letting |ā| = |a|. In certain interesting cases this norm satisfies condition (iii) above, and GD forms an arithmetical semigroup. The following are illustrations.

Firstly, if D is the ring Z of all rational integers, then it is clear that GD may be identified essentially with the semigroup Gz discussed above. A different example arises when D denotes the ring Z[√ — 1] of all Gaussian integers m+n√ – 1 (m,n Z). It is familiar that this ring forms a Euclidean domain if one assigns the norm m²+n² to the number m+n√ — 1. Since Z[√ — 1] has only the four units 1, –1, √ – 1, – √ – 1, one sees that

where r(n) denotes the total number of lattice points (a, b) (i.e., points (a, b) with integer components a, bin the Euclidean plane R². Thus GZ[√_1] forms an arithmetical semigroup.

For the sake of general interest, we note that the primes of GZ[√ – 1] are the associate classes in Z[√ – 1] of the numbers

(i) 1 + √ – 1

(ii) all rational primes p ≡ 3 (mod 4),

(iii) all factors a+b√ – 1 of rational primes p ≡ 1 (mod 4). For a proof of this statement, see Hardy and Wright [1], say.

Another well-known Euclidean domain is the domain Z[√2] of all real numbers of the form m+n√2 (m,n Z), in which m + n√2 is assigned the norm |m² —2n²|. This domain has infinitely many units; in fact, they are all the numbers of the form ±(1+√2)n (n√Zfinite for each x>0, and hence that GZ[√2] forms an arithmetical semigroup also. (See the book of Cohn [1], which in fact makes a study of general quadratic number domains. For the statement about units above, and the following remark, see for example Hardy and Wright [1].)

The primes of GZ[√2] have a classification similar to the one mentioned for GZ[√–1] above: they are the associate classes in Z[√2] of the numbers

(i) √2

(ii) all rational primes p ≡ ± 3 (mod 8),

(iii) all factors a+b√2 of rational primes p ≡ ±1 (mod 8).

Lastly, consider a polynomial ring F[t] in an indeterminate t over a field F. This is a familiar example of a Euclidean domain, in which one usually defines |f| = 2deg f for 0 ≠f F[t]. In the case when F is a finite Galois field GF(q), it is easy to see that GF[t] forms an arithmetical semi-group. In fact, one notes that

1.3. Example: Ideals in an algebraic number field. Although we shall not