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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.
Product Details
| ISBN-13: | 9780486169347 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 02/17/2015 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 352 |
| File size: | 24 MB |
| Note: | This product may take a few minutes to download. |
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Abstract Analytic Number Theory
By John KnopfmacherDover Publications, Inc.
Copyright © 1990 John KnopfmacherAll rights reserved.
ISBN: 978-0-486-16934-7
Contents
Preface to the Dover edition,Preface to the first edition,
Introduction,
Part I. ARITHMETICAL SEMIGROUPS AND ALGEBRAIC ENUMERATION PROBLEMS,
Chapter 1. Arithmetical semigroups,
Chapter 2. Arithmetical functions,
Chapter 3. Enumeration problems,
Part II. ARITHMETICAL SEMIGROUPS WITH ANALYTICAL PROPERTIES OF CLASSICAL TYPE,
Chapter 4. Semigroups satisfying axiom A,
Chapter 5. Asymptotic enumeration, and further "statistical" properties,
Chapter 6. The abstract prime number theorem,
Chapter 7. Fourier analysis of arithmetical functions,
Part III. ANALYTICAL PROPERTIES OF OTHER ARITHMETICAL SYSTEMS,
Chapter 8. Additive arithmetical semigroups,
Chapter 9. Arithmetical formations,
Appendix 1. Some unsolved questions,
Appendix 2. Values of p(n) AND s(n),
List of special symbols,
Bibliography,
Additional bibliography,
Index,
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
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
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 [member of] P,
(ii) |ab| = |a b| for all a,b [member of] G,
(iii) the total number NG(x) of elements a [member of] G of norm |a| [??] x 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 [member of] P of norm |p| [??] x 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, so that the number [bar.m] the greatest integer not exceeding x.
Although the function Nz(x) would be too trivial to mention if one were not interested in a wider arithmetical theory, the corresponding function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] remains mysterious to this day. The asymptotic behaviour of π(x) for large x forms the content of the Prime Number Theorem, which states that
π(x) ~ x/log x/log x as x [right arrow] ∞.
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 [member of] D are called associated if and only if a=bu for some unit u [member of] 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 [bar.a] of non-zero elements a [member of] D forms a commutative semigroup with identity under the multiplication operation [bar.a] x [bar.b] = [bar.ab]. In the case when D is a principal ideal domain, the content of the Unique Factorization Theorem for D is that the elements [bar.a] ≠ [bar.1] of GD admit unique factorization into powers of the classes [bar.p] of prime elements p [member of] 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 |[bar.a]| = |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 [v — 1] of all Gaussian integers m+n [??] – 1 (m,nZ). 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
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
where r(n) denotes the total number of lattice points (a, b) (i.e., points (a, b) with integer components a, b) on the circle x2 + y2 = n (n [??] 1) in 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 [equivalent to] 3 (mod 4),
(iii) all factors a+b [??] – 1 of rational primes p [equivalent to] 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 [member of] 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]??]Z). Nevertheless, it can be proved that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] finite 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 [equivalent to] ± 3 (mod 8),
(iii) all factors a+b][??]2 of rational primes p [equivalent to] ±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 [member of] F[t]. In the case whenF 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
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
1.3. Example: Ideals in an algebraic number field. Although we shall not presuppose a previous knowledge of algebraic number theory or of some of the concepts discussed in the next section, as indicated in the Introduction, we do feel it highly desirable to at least briefly outline some facts about the existence and general nature of various interesting examples of natural arithmetical semigroups in different fields of mathematics. The reader not familiar with some of the terms employed may perhaps still find it useful to glance generally at some of the examples motivating a study of abstract arithmetical semigroups, leaving a deeper consideration of individual systems to some later stage. In particular, the next few paragraphs are intended to be viewed in the spirit of the foregoing remarks.
Let K denote an algebraic number field, i.e., a finite extension of the rational field Q, and let D denote the ring of all algebraic integers contained in K. For example, D coincides respectively with the previously considered rings Z, Z [[??] — 1] or Z][??]2] according as K denotes Q, Q([??] – 1) or Q ([??]2), where Q ([??]m) denotes the set of all complex numbers of the form a+b]??]m (a,b [member of] Q). In the general case of an arbitrary algebraic number field K, the ring D is an integral domain which need not necessarily be a unique factorization domain. However, Dedekind showed that unique factorization can be "restored" in a sense, if one considers the set GK of all non-zero ideals in D (also called the ITL∫ITL ideals in K).
It is easy to see that GK forms a commutative semigroup with D itself as identity element, under the usual multiplication of ideals in a commutative ring. The sense in which unique factorization in D may be "restored" is given by the statement that every proper ideal I can be factorized uniquely into a product of powers of prime ideals. (For a domain D of the present type, a prime ideal is the same as a maximal ideal in D, and is irreducible (indecomposable) with respect to ideal multiplication.)
It is an interesting fact that the quotient ring D/I is finite in cardinal for each ideal I [member of] GK and that the definition |I| = card (D/I) provides a norm function on GK satisfying the conditions (i) to (iii) for an arithmetical semigroup. Moreover, the prime ideals in D all arise by factorizing the principal ideals (p) for all rational primes p. Thus (p) = P1P2 ... Pm, say, where Pi denotes a prime ideal, and since it can be shown that (p) has norm p[K:Q], it follows from the multiplicative property (ii) of the norm that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is finite for each x>0. (For an elementary discussion of ideals in an algebraic number field, see for example Pollard [1] or LeVeque [1].)
The asymptotic behaviour of the function πK(x) for large x forms the content of Landau's Prime Ideal Theorem:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];
this will be discussed again later on.
Before turning to the next section, it may be noted that, if the above domain D happens to be a principal ideal domain, then the previously- defined semigroup GD coincides essentially (i.e., is isomorphic) with the semigroup GK of ideals. Further, for special domains such as Z, Z [[??] – 1] and Z][??]2], the natural isomorphism [bar.a] (a) preserves norms. Thus the corresponding special arithmetical semigroups for these domains can all be subsumed under the study of the general arithmetical semigroups GK for arbitrary algebraic number fields K.
§ 2. Categories satisfying theorems of the Krull–Schmidt type
The following discussion is concerned mainly with certain specific classes of mathematical objects, such as groups, rings, topological spaces, and so on, together with the standard "direct product" operations and isomorphism relations appropriate to those classes. It is convenient, though admittedly not quite precise, to temporarily ignore the corresponding morphisms and refer to such classes of objects as categories. For example, in talking of the category G of all finite groups, one often thinks intuitively of the groups (or their isomorphism classes) alone, even though the importance of general homomorphisms is undeniable. Also, although abstract category theory will not be assumed or used here, the author personally believes that certain concepts, such as those of 'morphism' or 'functor' for example, may in due course become important in extending the type of arithmetical theory considered below. It is for these reasons that the term 'category' will be retained here, despite the slight abuse of language or lack of immediate necessity.
Now consider some category [??] which admits a "direct product" or "sum" operation × on its objects. Suppose that this operation × preserves [??]-isomorphisms, that it is commutative and associative up to [??]-isomorphism, and that [??] contains a "zero" object 0 (unique up to [??]-isomorphism) such that A × 0 [congruent to] A for all objects A in [??]. Then suppose that a theorem of the Krull–Schmidt type is valid for [??], i.e., suppose that every object A [not congruent to] 0 can be expressed as a finite ×-product A [congruent to] P1 × P2 × ... × Pm of objects Pi [not congruent to] 0 that are indecomposable with respect to ×, in a way that is unique up to permutation of terms and [??]-isomorphism. In most natural situations at least, one may reformulate these conditions on [??] by stating that the various isomorphism classes [bar.A] of objects A in [??] form a set G[??] that is
(i) a commutative semigroup with identity with respect to the multiplication operation [bar.A] × [bar.B] = [bar.A × B],
(ii) a semigroup with the unique factorization property with respect to the isomorphism classes of the indecomposable objects in [??].
For this reason, one may call the [??]-isomorphism classes [bar.p] of indecomposable objects P the primes of [??] or G[??].
It has always been obvious that categories with properties of the above kind are in some senses formally remniscent of the multiplicative semigroup Gz of positive integers. However, less attention appears to have been paid to the fact that, in many interesting cases (some of which are illustrated below), the category [??] also admits a real-valued norm function on objects which is invariant under [??]-isomorphism and has the following properties:
(i) |0| = 1, |P|>1 for every indecomposable object P,
(ii) |A × B| = |A B| for all objects A, B,
(iii) the total number of N[??](x) of [??]-isomorphism classes of objects A of norm |A|[??]x is finite, for each real x0.
Obviously, in such circumstances, the definition |bar.A| = |A| provides a norm function on G[??] satisfying the required conditions for an arithmetical semigroup. For these reasons, a category [??] with such further properties may be called an arithmetical category.
(Continues...)
Excerpted from Abstract Analytic Number Theory by John Knopfmacher. Copyright © 1990 John Knopfmacher. Excerpted by permission of Dover Publications, Inc..
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