Introduction to Arithmetical Functions

372 Pages·1986·20.5 MB·English

by  Paul J. McCarthy (auth.)

Universitext
Editors
F.W. Gehring
P.R. Halmos
C.C. Moore
Universitext
Editors: F.W. Gehring, P.R. Halmos. c.c. Moore
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Chern: Complex Manifolds Without Potential Theory
Chorin/Marsden: A Mathematical Introduction to Fluid Mechanics
Cohn: A Classical Invitation to Algebraic Numbers and Class Fields
Curtis: Matrix Groups, 2nd ed.
van Dalen: Logic and Structure
Devlin: Fundamentals of Contemporary Set Theory
Edwards: A Formal Background to Mathematics I alb
Edwards.: A Formal Background to Higher Mathematics II alb
Endler: Valuation Theory
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Gardiner: A First Course in Group Theory
Godbillon: Dynamical Systems on Surfaces
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Kelly/Matthews: The Non-Euclidean, The Hyperbolic Plane
Kostrikin: Introduction to Algebra
Luecking/Rubel: Complex Analysis: A Functional Analysis Approach
Lu: Singularity Theory and an Introduction to Catastrophe Theory
Marcus: Number Fields
McCarthy: Introduction to Arithmetical Functions
Meyer: Essential Mathematics for Applied Fields
Moise: Introductory Problem Course in Analysis and Topology
0ksendal: Stochastic Differential Equations
Porter/Woods: Extensions of Hausdorff Spaces
Rees: Notes on Geometry
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Schreiber: Differential Fomls
Smorynski: Self-Reference and Modal Logic
Stanish:: The Mathematical Theory of Turbulence
Stroock: An Introduction to the Theory of Large Deviations
Tolle: Optimization Methods
Paul J. McCarthy
Introduction to
Arithmetical Functions
Springer-Verlag
New York Berlin Heidelberg Tokyo
Paul J. McCarthy
Department of Mathematics
University of Kansas
Lawrence, KS 66045
U.S.A.
AMS Classifications: 10-01, lOA20, lOA21 , lOH25
Library of Congress Cataloging-in-Publication Data
McCarthy, Paul J. (Paul Joseph)
Introduction to arithmetical functions.
(Universitext)
Bibliography: p.
Includes index.
l. Arithmetic functions. I. Title.
QA245.M36 1985 512'.7 85-26068
With 6 illustrations.
© 1986 by Springer-Verlag New York Inc.
Reprint ofthe original edition 1986
All rights reserved. No part of this book may be translated or reproduced in any form
without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New
York 10010, U.S.A.
987 6 5 4 3 2 1
ISBN-13: 978-0-387-96262-7 e-ISBN-13: 978-1-4613-8620-9
DOl: 10.1007/978-1-4613-8620-9
Preface
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.
Each chapter is followed by notes which are bibliographical in nature,
and only incidentally historical. My purpose in the notes is to point
out sources of results. Number theory, and the theory of arithmetical
functions in particular, is rife with rediscovery, so I hope the reader
will not be too harsh with me if I fail to pin down the truly first
source of some result. Perhaps this book will help reduce the rate
of rediscovery.
There are more than four hundred exercises. They form an
essential part of my development of the subject, and during any serious
reading of the book some time must be spent thinking about the exercises.
I assume that the reader is familiar with calculus, including
infinite series, and has the maturity gained from completing several
mathematics courses at the college level. A first course in the
theory of numbers provides more than enough background in number
theory. In fact, only a few things from such a course are used, such
as ele~entary properties of congruences and the unique factorization
theorem.
Table of Contents
Chapter l. Multiplicative Functions
Chapter 2. Ramanujan Sums 70
Chapter 3. Counting Solutions of Congruences 114
Chapter 4. Generalizations of Dirichlet Convolution 149
Chapter 5. Dirichlet Series and Generating Functions 184
Chapter 6. Asymptotic Properties of Arithmetical Functions 255
Chapter 7. Generalized Arithmetical Functions 293
References 333
Bibliography 334
Index 361
Chapter 1
Multiplicative Functions
Throughout this book rand 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.
An arithmetical function is a complex-valued function defined on the
set of positive integers.
Although examples of such functions can be defined in a completely
arbitrary manner, the most interesting ones are those that arise from
some arithmetical consideration. Our first examples certainly do arise in
this way. The Euler function ¢ is defined by
¢en) the number of integers x such that < x < n and ex, n) 1.
If k is a nonnegative integer, the function Ok is defined by
Ok en) ; the sum of the kth powers of the divisors of n.
The term "divisor" always means "positive divisor." We can express Ok en)
by using the sigma-notation:
for all n.
2
In particular,
o(n) the sum of the divisors of n,
and
T (n) the number of divisors of n.
If k is a nonnegative integer, the function sk is defined by
k
n The function is called the zeta function:
Sen) for all n.
There are several useful binary operations on the set of arithmetical
functions. If f and g are arithmetical functions their Sum f + g
and product fg are defined in the usual way:
(f + g) (n) f(n)+g(n) for all n
and'
(f g) (n) f (n) g (n) for all n.
The (Dirichlet) convolution f * g of f and g is defined by
L.
(f ~, g) (n) f(d)g(n/d) for all n.
din
For example, ok = sk * S .
Addition and multiplication of arithmetical functions have all the
usual properties of commutativity, associativity, distributivity, etc.
3
Proposition 1.1. If f, g and h are arithmetical functions then
* *
(i) f g ; g f
(ii) (f * g) * h f * (g * h)
(iii) f * (g + h) f * g + f * h
Proof. (i) follows from the fact that as d runs over all the
divisors of n , so does n/d. Nmv, for all n
(f * (g * h)) (n) ; ~ fed) ~ g(e)h(n!de) ,
din
and if D de this is equal to
~ ~ f(D/e)g(e))h(n/D)
Din elD
«f * *
g) h)(n).
This proves (ii). As for (iii), for all n,
(f * (g + h)) (n) ~ fed) (g(n/d) + hen/d))
din
~ f(d)g(n/d) + ~ f(d)h(n/d)
din din
(f * g)(n) + (f * h)(n)
(f * g + f * h ) (n). 0
Thus, in the language of abstract algebra, the set of arithmetical
functions, together with the binary operations of addition and convolution,