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1986 | Buch

Multiplicative Functions Kapitel lesen Erstes Kapitel lesen

Introduction to Arithmetical Functions

verfasst von: Paul J. McCarthy

Verlag: Springer New York

Buchreihe : Universitext

Enthalten in: Professional Book Archive

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Über dieses Buch

The theory of arithmetical functions has always been one of the more active parts of the theory of numbers. The large number of papers in the bibliography, most of which were written in the last forty years, attests to its popularity. Most textbooks on the theory of numbers contain some information on arithmetical functions, usually results which are classical. My purpose is to carry the reader beyond the point at which the textbooks abandon the subject. In each chapter there are some results which can be described as contemporary, and in some chapters this is true of almost all the material. This is an introduction to the subject, not a treatise. It should not be expected that it covers every topic in the theory of arithmetical functions. The bibliography is a list of papers related to the topics that are covered, and it is at least a good approximation to a complete list within the limits I have set for myself. In the case of some of the topics omitted from or slighted in the book, I cite expository papers on those topics.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Multiplicative Functions
Abstract
Throughout this book r and n, and certain other letters, are integer variables. Without exception r is restricted to the positive integers. Unless it is stated to the contrary, the same is true of n and other integer variables. However, on some occasions n and other integer variables will be allowed to have negative and zero values. On such occasions it will be stated explicitly that this is the case.
Paul J. McCarthy
Chapter 2. Ramanujan Sums
Abstract
If a and b are integers let
$$ e(a,b) = {e^{{\frac{{2\pi ia}}{b}}}} $$
Let n be an integer, positive, negative or zero, and let r be a positive integer. Consider the sum
$$ c(n,r) = \sum\limits_{{(x,r) = 1}} {e(nx,r)} $$
Usually, the sum is taken over all × such that 1 ≤ x ≤ r and (x, r) = 1, but it could be over any reduced residue system (mod r). This is because, if x ≡ x′ (mod r) then e(nx, r) = e(nx′, r). The sum c(n,r) is called a Ramanujan sum. For fixed r, and with n restricted to the positive integers, we obtain an arithmetical function c(·, r). Some authors devote this function by cr, so that cr(n) = c(n, r)
Paul J. McCarthy
Chapter 3. Counting Solutions of Congruences
Abstract
In this chapter we shall use the results obtained in the preceding chapter to count solutions of certain linear and other congruences in s unknowns. By a solution of a congruence, with modulus r, we mean a solution (mod r), i.e., an ordered s-tuple of integers (x1,…, xs) that satisfies the congruence, with two s-tuples (x1,…, xs) and \( \langle x_1^{'},...,x_s^{'}\rangle \) that satisfy the congruence counted as the same solution if and only if xi ≡ x1′. (mod r) for i = 1,…, s.
Paul J. McCarthy
Chapter 4. Generalizations of Dirichlet Convolution
Abstract
Let K be a complex-valued function on the set of all ordered pairs <n,d> where n is a positive integer and d is a divisor of n. If f and g are arithmetical functions, their K-convolution, f *K g, is defined by
$$ (f{*_K}g)(n) = \sum\limits_{{d\left| n \right.}} K (n,d)f(d)g(n/d)\quad {\text{for}}\,{\text{all}}\,{\text{n}} $$
Paul J. McCarthy
Chapter 5. Dirichlet Series and Generating Functions
Abstract
A series of the form
$$ \sum\limits_{{n - 1}}^{\infty } {\frac{{f(n)}}{{{n^s}}}} $$
(*)
where f is an arithmetical function and s is a real variable, is called a Dirichlet series. It will be called the Dirichlet series of f. There exist Dirichlet series such that for all values of s, the series does not converge absolutely (see Exercise 5.1). If the Dirichlet series of f does converge absolutely for some values of s then for those values of s the series determines a function which, as we shall see, serves as a generating function of f.
Paul J. McCarthy
Chapter 6. Asymptotic Properties of Arithmetical Functions
Abstract
Let us begin with an example. The object is to describe in some meaningful way the behavior of
$$ \sum\limits_{{n \leqslant x}} {\tfrac{1}{n}} $$
as a function of the real variable x, for large x. To do this we need the following information.
Paul J. McCarthy
Chapter 7. Generalized Arithmetical Functions
Abstract
Many of the properties of arithmetical functions, especially inversion properties and arithmetical identities, hold in a much more general setting than we have used in this book. In this chapter we shall introduce the reader to this general setting, look at various examples, and obtain some general results that can be applied in the special situations contained in the examples.
Paul J. McCarthy
Backmatter
Metadaten
Titel
Introduction to Arithmetical Functions
verfasst von
Paul J. McCarthy
Copyright-Jahr
1986
Verlag
Springer New York
Electronic ISBN
978-1-4613-8620-9
Print ISBN
978-0-387-96262-7
DOI
https://doi.org/10.1007/978-1-4613-8620-9