Arithmetical Functions

387 Pages·1994·4.1 MB·English

by  Wolfgang Schwarz

131 Algebraic, extrcmal & metric combinatorics, M-M. DEZA, P FRANKL & I.G. ROSENBERG (eds)
132 Whitehead groups of finite groups, ROBERT OLIVER
133 Linear algebraic monoids, MOHAN S. PUTCHA
134 Number theory and dynamical systems, M. DODSON & J. VICKERS (eds)
135 Operator algebras and applications, 1, D. EVANS & M. TAKESAKI (eds)
136 Operator algebras and applications, 2, D. EVANS & M.'i'AKESAKI (eds)
137 Analysis at Urbana, I, E. BERKSON, T. PECK, & J. UHL (eds)
138 Analysis at Urbana, II, E. BERKSON, T. PECK, & J. UIIL (eds)
139 Advances in homotopy theory, S. SALAMON, B STEER & W. SUTHERLAND (eds)
140 Geometric aspects of Banach spaces, E M. PEINADOR and A. RODES (eds)
141 Surveys in combinatorics 1989, J. SIEMONS (ed)
142 The geometry of jet bundles, D J. SAUNDERS
143 The ergodic theory of discrete groups, PETER J. NICIHOLLS
144 Introduction to uniform spaces, I.M. JAMES
145 Ilomological questions in local algebra, JAN R ST'ROOKER
146 Cohen-Macaulay modules over Cohcn-Macaulay rings, Y. YOSHINO
147 Continuous and discrete modules, S.H. MOIIAMED & B J. MULLER
148 Helices and vector bundles, A N. RUDAKOV et at
149 Solitons nonlinear evolution equations & inverse scattering, M. ABLOWI'IZ & P. CI.ARKSON
150 Geometry of low-dimensional manifolds 1, S. DONALDSON & C.B. THOMAS (eds)
151 Geometry of low-dimensional manifolds 2, S. DONALDSON & C B. THOMAS (eds)
152 Oligomorphic permutation groups, P. CAMERON
153 L-functions and arithmetic, J. COATES & M.J. TAYLOR (eds)
154 Number theory and cryptography, J. LOXTON (ed)
155 Classification theories of polarized varieties, TAKAO I.1JJITA
156 Twistors in mathematics and physics, 'I'.N. BAILEY & R J. BASTON (eds)
157 Analytic pro-p groups, J.D. DIXON, M.P.F DU SAUTOY, A. MANN & D. SEGAL
158 Geometry of Banach spaces, P F.X. MU)LLER & W. SCIIACIIERMAYER (eds)
159 Groups St Andrews 1989 volume 1, C.M CAMPBELL & E F. ROBERTSON (eds)
160 Groups St Andrews 1989 volume 2, C M CAMPBELL. & E F. ROBERTSON (eds)
161 Lectures on block theory, BURKHARI) KULSHAMMER
162 Harmonic analysis and representation theory for groups acting on homogeneous trees,
A. FIGA-T'ALAMANCA & C. NEBBIA
163 Topics in varieties of group representations, S.M. VOVSI
164 Quasi-symmetric designs, M.S. SitRIKANDE & S.S. SANE
165 Groups, combinatoncs & geometry, M.W. LIEBECK & J. SAXL (eds)
166 Surveys in combinatorics, 1991, A.D. KEEDWELL (ed)
167 Stochastic analysis, M.T. BARLOW & N.H. BINGHAM (eds)
168 Representations of algebras, H. TACHIKAWA & S. BRENNER (eds)
169 Boolean function complexity, M.S. PATERSON (ed)
170 Manifolds with singularities and the Adams-Novikov spectral sequence, B. BOTVINNIK
171 Squares, A R. RAJWADE
172 Algebraic varieties, GEORGE R. KEMPF
173 Discrete groups and geometry, W.J. HARVEY & C. MACI.ACIILAN (eds)
174 Lectures on mechanics, J.E. MARSDEN
175 Adams memorial symposium on algebraic topology 1, N. RAY & G. WALKER (eds)
176 Adams memorial symposium on algebraic topology 2, N RAY & G. WALKER (eds)
177 Applications of categories in computer science, M.P FOURMAN, P.T. JOHNSTONE,
& A.M. PITT'S (eds)
178 Lower K- and L-theory, A RANICKI
179 Complex projective geometry, G. ELLINGSRUD, C. PFSKINE, G. SACCHIERO
& S.A STROMME (eds)
180 Lectures on crgodic theory and Pcsin theory on compact manifolds, M POLLICOTT
181 Geometric group theory I, G A. NIBLO & M A ROLLER (eds)
182 Geometric group theory II, G.A. NIBLO & M A. ROLLER (cds)
183 Shintani zeta functions, A YUKIE
184 Arithmetical functions, W. SCHWARZ & J SPILKER
185 Representations of solvable groups, O. MANZ & T.R. WOLF
186 Complexity: knots, colourings and counting, D J.A. WELSH
187 Surveys in combinatorics, 1993, K. WALKER (ed)
189 Locally presentable and accessible categories, J. ADAMEK & J. ROSICKY
190 Polynomial invariants of finite groups, DJ BENSON
191 Finite geometry and combinatorics, F DE CLERCK el at
192 Symplectic geometry, D. SALAMON (cd)
197 Two-dimensional homotopy and combinatorial group theory, C. HOG-ANGELONI
W. METZLER & A J. SIERADSKI (eds)
198 The algebraic characterization of geometric 4-manifolds, J.A. HILLMAN
London Mathematical Society Lecture Note Series. 184
Arithmetical Functions
An Introduction to Elementary and Analytic Properties of
Arithmetic Functions and to some of their Almost-Periodic
Properties
Wolfgang Schwarz
Johann Wolfgang Goethe-Universitt t, Frankfurt am Main
Jurgen Spilker
Freiburg im Breisgau
CAMBRIDGE
UNIVERSITY PRESS
Published by the Press Syndicate of the University of Cambridge
The Pitt Building, Trumpington Street, Cambridge CB2 1RP
40 West 20th Street, New York, NY 10011-4211, USA
10 Stamford Road, Oakleigh, Melbourne 3166, Australia
© Cambridge University Press 1994
First published 1994
Printed in Great Britain at the University Press, Cambridge
British Library cataloguing in publication data available
Library of Congress cataloguing in publication data available
ISBN 0 521 42725 8
To OUR Wives DORIS and HELGA

Contents
..
preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
..
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Chapter I Tools from Number Theory
. . . . . . . . . . . . 1
I.I. Partial Summation . . . . . . . . . . . . . . . . . . . . . 2
1.2. Arithmetical Functions, Convolution, Mdbius Inversion Formula 4
1.3. Periodic Functions, Even Functions, Ramanujan Sums 15
1.4. The Turin-Kubillus Inequality . . . . . . . . . . . . . . 19
I.S. Generating Functions, Dirichlet Series . . . . . . . . . . 25
1.6. Some Results on Prime Numbers . . . . . . . . . . . . . 31
1.7. Characters, L-Functions, Primes in Arithmetic Progressions 3S
1.8. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Photographs 43
. . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter II Mean-Value Theorems and Multiplicative Functions, I 4S
11.1. Motivation 46
. . . . . . . . . . . . . . . . . . . . . . . . . .
11.2. Elementary Mean-Value Theorems (Wlntner, Axer) 49
. .
11.3. Estimates for Sums over Multiplicative Functions (Rankin's
Trick) . . . . . . . . . . . . . . . . . . . . . . . . . 56
11.4. Wirsing's Mean-Value Theorem for Sums over Non-Negative
Multiplicative Functions . . . . . . . . . . . . . . . 65
II.S. The Theorem of G. Halasz on Mean-Values of Complex-
Valued Multiplicative Functions . . . . . . . . . . 76
11.6. The Theorem of Daboussi and Delange on the Fourier-Coef-
ficients of Multiplicative Functions 78
. . . . . . .
11.7. Application of the Daboussi-Delange Theorem to a Problem
of Uniform Distribution . . . . . . . . . . . . . . . 81
11.8. The Theorem of Saffari and Daboussi, I. 82
. . . . . . . . .
11.9. Daboussi's Elementary Proof of the Prime Number Theorem 85
11.10. Mohan Nair's Elementary Method in Prime Number Theory 91
- vii -
Contents
11.11. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Chapter III Related Arithmetical Functions . . . . . . . . . . 97
111.1. Introduction, Motivation 98
. . . . . . . . . . . . . . . . .
111.2. Main Results . . . . . . . . . . . . . . . . . . . . . . . . 101
111.3. Lemmata, Proof of Theorem 2.3 104
. . . . . . . . . . . .
111.4. Applications . . . . . . . . . . . . . . . . . . . . . . . . . 110
III.S. On a Theorem of L. Lucht
. . . . . . . . . . . . . . . . 115
111.6. The Theorem of Saffari and Daboussi, II
. . . . . . . . 117
111.7. Application to Almost-Periodic Functions . . . . . . . 118
111.8. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Chapter IV Uniformly Almost-Periodic Arithmetical
Functions . . . . . . . . . . . . . . . . . . . . . . . 123
IV.1. Even and Periodic Arithmetical Functions . . . . . . . 124
IV.2. Simple Properties . . . . . . . . . . . . . . . . . . . . . 133
IV.3. Limit Distributions . . . . . . . . . . . . . . . . . . . . 139
IV.4. Gelfand's Theory: Maximal Ideal Spaces . . . . . . . 142
IV.4.A. The maximal ideal space 0B of ,$U 142
IV.4.B. The maximal ideal space 0., of Bu 147
IV.S. Application of Tietze's Extension Theorem . . . . . 155
IV.6. Integration of Uniformly Almost-Even Functions IS6
. .
IV.7. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Chapter V Ramanujan Expansions of Functions in 8" . . . . 165
V.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 166
V.2. Equivalence of Theorems 1.2, 1.3, 1.4, 1.S . . . . . . . . 168
V.3. Some Lemmata . . . . . . . . . . . . . . . . . . . . . . 171
V.4. Proof of Theorem 1.5 . . . . . . . . . . . . . . . . . . . 175
V.S. Proof of Lemmas 3.4 and 3.5 . . . . . . . . . . . . . . . 178
V.6. Exercises 184
. . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter VI Almost-Periodic and Almost-Even Arithmetical
Functions . . . . . . . . . .. . . . . . . . ... . . . . 185
VI.1. Besicovich Norm, Spaces of Almost Periodic Functions 186
VI.2. Some Properties of Spaces of q-Almost-Periodic Functions 197
- viii -
Contents
VI.3. Parseval's Equation 206
. . . . . . . . . . . . . . . . . . .
VI.4. A Second Proof for Parseval's Formula 208
. . . . . . .
VI.S. An Approximation for Functions in S1 . . . . . . . 210
VI.6. Limit Distributions of Arithmetical Functions . . . 212
VI.7. Arithmetical Applications
. . . . . . . . . . . . . . 21S
VI.7. A. Mean-Values, Limit Distributions . . . . 215
VI.7.B. Applications to Power-Series with Multiplicative
Coefficients . . . . . . . . . . . . . . . . . 218
VI.7.C. Power Series Bounded on the Negative Real Axis 221
VI.8. A 2 q - Criterion . . . . . . . . . . . . . . . . . . . . . . 224
VI.9. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Photographs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Chapter VII The Theorems of Elliott and Daboussi 233
VII.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 234
VII.2. Multiplicative Functions with Mean-Value M(f) * 0, Satis-
fying II f 112 < CO . . . . . . . . . . . . . . . . . . . . 239
21
VII.3. Criteria for Multiplicative Functions to Belong to 243
VII.4. Criteria for Multiplicative Functions to Belong to 2q 251
VII.S. Multiplicative Functions in Aq with Mean-Value M(f) $ 0 257
VII.6. Multiplicative Functions in ,4" with Non-Void Spectrum 261
VII.7. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 266
Chapter VIII Ramanujan Expansions 269
. . . . . . . . . . . . . . .
VIII.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 270
VIII.2. Wintner's Criterion . . . . . . . . . . . . . . . . . . . . 271
VIII.3. Mean-Value Formulae for Multiplicative Functions 276
VIII.4. Formulae for Ramanujan Coefficients 280
. . . . . . . .
VIII.S. Pointwise Convergence of Ramanujan Expansions 284
VIII.6. Still Another Proof for Parseval's Equation 289
. . . .
VIII.7. Additive Functions . . . . . . . . . . . . . . . . . . . . 291
VIII. 8. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 291
Contents
Chapter IX Mean-Value Theorems and Multiplicative Functions, II 293
IX.1. On Wirsing's Mean-Value Theorem 294
. . . . . . . . . .
IX.2. Proof of Theorem 1.4 . . . . . . . . . . . . . . . . . 298
IX.3. The Mean-Value Theorem of Gabor Halasz 303
. . . .
IX.4. Proof of Proposition 3.3 . . . . . . . . . . . . . . . . . 309
IX.S. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 311
Photographs . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
A.I. The Stone-Weierstrass Theorem, Tietze's Theorem . . 315
A.2. Elementary Theory of Hilbert Space . . . . . . . . . . 316
A.3. Integration . . . . . . . . . . . . . . . . . . . . . . . 319
A.4. Tauberian Theorems (Hardy-Littlewood-Karamata, 321
Landau-Ikehara)
. . . . . . . . . . . . . . .
A.S. The Continuity Theorem for Characteristic Functions 323
A.6. Gelfand's Theory of Commutative Banach Algebras 325
A.7. Infinite Products . . . . . . . . . . . . . . . . . . . 327
A. 8. The Large Sieve . . . . . . . . . . . . . . . . . . . 329
A.9. Dirichlet Series . . . . . . . . . . . . . . . . . . . . 331
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
Author Index 353
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Subject Index 357
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Photographs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36S
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . 367