Mathematical Surveys and Monographs Volume 213 Beurling Generalized Numbers Harold G. Diamond Wen-Bin Zhang (Cheung Man Ping) American Mathematical Society https://doi.org/10.1090//surv/213 Beurling Generalized Numbers Mathematical Surveys and Monographs Volume 213 Beurling Generalized Numbers Harold G. Diamond Wen-Bin Zhang (Cheung Man Ping) American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Robert Guralnick Benjamin Sudakov Michael A. Singer, Chair Constantin Teleman MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 11N80. For additional information and updates on this book, visit www.ams.org/bookpages/surv-213 Library of Congress Cataloging-in-Publication Data Names: Diamond, Harold G., 1940–. Zhang, Wen-Bin (Cheung, Man Ping), 1940– . Title: Beurling generalized numbers / Harold G. Diamond, Wen-Bin Zhang (Cheung Man Ping). Description: Providence, Rhode Island : American Mathematical Society, [2016] | Series: Mathe- matical surveys and monographs ; volume 213 | Includes bibliographical references and index. Identifiers: LCCN 2016022110 | ISBN 9781470430450 (alk. paper) Subjects: LCSH: Numbers, Prime. | Numbers, Real. | Riemann hypothesis. | AMS: Number theory – Multiplicative number theory – Generalized primes and integers. msc Classification: LCC QA246 .D5292 2016 | DDC 512/.2–dc23 LC record available at https://lccn.loc.gov/2016022110 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. 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Visit the AMS home page at http://www.ams.org/ 10987654321 212019181716 Dedicated to our wives, Nancy Diamond and Kun-Ming Luo Zhang, and to the memory of our friend and colleague, Paul T. Bateman Contents Preface xi Chapter 1. Overview 1 1.1. Some questions about primes 1 1.2. The cast 2 1.3. Examples 4 1.4. π, Π, and an extended notion of g-numbers 6 1.5. Notes 7 Chapter 2. Analytic Machinery 9 2.1. A function class 9 2.2. Measures 9 2.3. Mellin transforms 10 2.4. Norms 11 2.5. Convergence 11 2.6. Convolution of measures 12 2.7. Convolution of functions 15 2.8. The L and T operators 16 2.9. Notes 18 Chapter 3. dN as an Exponential and Chebyshev’s Identity 19 3.1. Goals and plan 19 3.2. Power series in measures 21 3.3. Inverses 22 3.4. The exponential on V 23 3.5. Three equivalent formulas 27 3.6. Notes 28 Chapter 4. Upper and Lower Estimates of N(x) 29 4.1. Normalization and restriction 29 4.2. O-log density 30 4.3. Lower log density 33 4.4. An example with infinite residue but 0 lower log density 36 4.5. Extreme thinness is inherited 37 4.6. Regular growth 39 4.7. Notes 40 Chapter 5. Mertens’ Formulas and Logarithmic Density 41 5.1. Introduction 41 5.2. Logarithmic density 41 5.3. The Hardy-Littlewood-Karamata Theorem 43 vii viii CONTENTS 5.4. Mertens’ sum formula 45 5.5. Mertens’ product formula 46 5.6. A remark on γ 47 5.7. An equivalent form and proof of “only if” 48 5.8. Tauber’s Theorem and conclusion of the argument 49 5.9. Notes 51 Chapter 6. O-Density of g-integers 53 6.1. Non-relation of log-density and O-density 53 6.2. O-Criteria for O-density 56 6.3. Sharper criteria for O-density 60 6.4. Notes 62 Chapter 7. Density of g-integers 63 7.1. Densities and right hand residues 63 7.2. Axer’s Theorem 63 7.3. Criteria for density 65 7.4. An L1 criterion for density 69 7.5. Estimates of N(x) with an error term 72 7.6. Notes 76 Chapter 8. Simple Estimates of π(x) 77 8.1. Unboundedness of π(x) 77 8.2. Can there be as many primes as integers? 78 8.3. π(x) estimates via regular growth 79 8.4. Lower bounds for 1/pi via lower log-density 81 8.5. Notes 81 Chapter 9. Chebyshev Bounds – Elementary Theory 83 9.1. Introduction 83 9.2. Chebyshev bounds for natural primes 83 9.3. An auxiliary function 87 9.4. Chebyshev bounds for g-primes 90 9.5. A failure of Chebyshev bounds 95 9.6. Notes 98 Chapter 10. Wiener-Ikehara Tauberian Theorems 99 10.1. Introduction 99 10.2. Wiener-Ikehara Theorems 99 10.3. The Fej´er kernel 101 10.4. Proof of the Wiener-Ikehara Theorems 104 10.5. A W-I oscillatory example 108 10.6. Notes 110 Chapter 11. Chebyshev Bounds – Analytic Methods 111 11.1. Introduction 111 11.2. Wiener-Ikehara setup 112 11.3. A first decomposition 113 11.4. Further decomposition of I2,σ(y) 114 11.5. Chebyshev bounds 116 CONTENTS ix 11.6. Notes 117 Chapter 12. Optimality of a Chebyshev Bound 119 12.1. Introduction 119 12.2. The g-prime system PB 119 12.3. Chebyshev bounds and the zeta function ζB(s) 122 12.4. The counting function NB(x) 124 12.5. Fundamental estimates 127 12.6. Proof of the Optimality Theorem 129 12.7. Notes 131 Chapter 13. Beurling’s PNT 133 13.1. Introduction 133 13.2. A lower bound for |ζ(σ + it)| 133 13.3. Nonvanishing of ζ(1 + it) 135 13.4. An L1 condition and conclusion of the proof 137 13.5. Optimality – a continuous example 139 13.6. Optimality – a discrete example 144 13.7. Notes 148 Chapter 14. Equivalences to the PNT 151 14.1. Introduction 151 14.2. Implications 151 14.3. Sharp Mertens relation and the PNT 152 14.4. Optimality of the sharp Mertens theorem 154 14.5. Implications between M(x)=o(x) and m(x)=o(1) 155 14.6. Connections of the PNT with M(x)=o(x) 156 14.7. Sharp Mertens relation and m(x)=o(1) 158 14.8. Notes 160 Chapter 15. Kahane’s PNT 161 15.1. Introduction 161 15.2. Zeros of the zeta function 162 15.3. A lower bound for |ζ(σ + it)| 165 15.4. A Schwartz function and Poisson summation 166 15.5. Estimating the sum of a series by an improper integral 170 15.6. Conclusion of the proof 172 15.7. Notes 174 Chapter 16. PNT with Remainder 175 16.1. Introduction 175 16.2. Two general lemmas 176 16.3. A Nyman type remainder term 179 16.4. A dlVP-type remainder term 186 16.5. Notes 192 Chapter 17. Optimality of the dlVP Remainder Term 195 17.1. Background 195 17.2. Discrete random approximation 196 17.3. Generalized primes satisfying the Riemann Hypothesis 204 xCONTENTS 17.4. Generalized primes with large oscillation 208 17.5. Properties of G(z) 209 17.6. Representation of log G(z) as a Mellin transform 210 17.7. A template zeta function 214 17.8. Asymptotics of NB(x) 218 17.9. Asymptotics of ψB(x) 222 17.10. Normalization and hybrid 226 17.11. Notes 227 Chapter 18. The Dickman and Buchstab Functions 229 18.1. Introduction 229 18.2. The ψ(x, y) function 230 18.3. The φ(x, y) function 233 18.4. A Beurling version of ψ(x, y) 234 18.5. G-numbers with primes from an interval 235 18.6. Other relations 237 18.7. Notes 238 Bibliography 239 Index 243 Preface Generalized numbers are a multiplicative structure introduced by A. Beurling [Be37] in 1937 to investigate the degree to which prime number theory is indepen- dent of the additive properties of the natural numbers. Beyond their own interest, the results and techniques of this theory apply to several other systems having the character of prime numbers and integers. Indeed, such ideas occurred already in a 1903 paper of E. Landau [La03] proving the prime number theorem for ideals of algebraic number fields. We shall introduce and use continuous (!) analogues of generalized (briefly: g-) numbers. As another applica- tion, these distributions provide an attractive path to the theories of Dickman and Buchstab for integers whose prime factors lie only in restricted ranges. A central question that we shall examine is the following: if a sequence of g-integers is generated by a sequence of g-primes, and if one of the collections is “reasonably near” its classical counterpart, does the other collection also have this property? This monograph does not examine all facets of g-number theory; some interesting topics that are largely ignored include probabilistic theory, oscillatory counting functions, and collections of primes and integers that are unusually dense or sparse.