Analytic. Number Theory. Exploring the Anatomy of Integers. Jean-Marie. De Koninck. Florian Luca. ffk li? Graduate Studies.

Transcription

1 Analytic Number Theory Exploring the Anatomy of Integers Jean-Marie Florian Luca De Koninck Graduate Studies in Mathematics Volume 134 ffk li? American Mathematical Society Providence, Rhode Island

2 Preface ix Notation xiii Frequently Used Functions xvii Chapter 1. Preliminary Notions Approximating a sum by an integral The Euler-MacLaurin formula The Abel summation formula Stieltjes integrals Slowly oscillating functions Combinatorial results The Chinese Remainder Theorem The density of a set of integers The Stirling formula Basic inequalities 13 Problems on Chapter 1 15 Chapter 2. Prime Numbers and Their Properties Prime numbers and their polynomial representations There exist infinitely many primes A first glimpse at the size of ir(x) Fermat numbers A better lower bound for 7r(a;) 24 iii

3 iv 2.6. The Chebyshev estimates The Bertrand Postulate The distance between consecutive primes Mersenne primes Conjectures on the distribution of prime numbers 33 Problems on Chapter 2 36 Chapter 3. The Riemann Zeta Function The definition of the Riemann Zeta Function Extending the Zeta Function to the half-plane cr > The derivative of the Riemann Zeta Function The zeros of the Zeta Function Euler's estimate <(2) tt2/6 = 45 Problems on Chapter 3 48 Chapter 4. Setting the Stage for the Proof of the Prime Number Theorem Key functions related to the Prime Number Theorem A closer analysis of the functions 0(x) and i)j(x) Useful estimates The Mertens estimate The Mobius function The divisor function 58 Problems on Chapter 4 60 Chapter 5. The Proof of the Prime Number Theorem A theorem of D. J. Newman An application of Newman's theorem The proof of the Prime Number Theorem A review of the proof of the Prime Number Theorem The Riemann Hypothesis and the Prime Number Theorem Useful estimates involving primes Elementary proofs of the Prime Number Theorem 72 Problems on Chapter 5 72 Chapter 6. The Global Behavior of Arithmetic Functions Dirichlet series and arithmetic functions The uniqueness of representation of a Dirichlet series 77

4 v 6.3. Multiplicative functions Generating functions and Dirichlet products Wintner's theorem Additive functions The average orders of w(n) and f2(n) The average order of an additive function The Erdos-Wintner theorem 88 Problems on Chapter 6 89 Chapter 7. The Local Behavior of Arithmetic Functions The normal order of an arithmetic function The Turan-Kubilius inequality Maximal order of the divisor function An upper bound for d(n) Asymptotic densities Perfect numbers Sierpiriski, Riesel, and Romanov Some open problems of an elementary nature 108 Problems on Chapter Chapter 8. The Fascinating Euler Function The Euler function Elementary properties of the Euler function The average order of the Euler function When is <j)(n)a(n) a square? The distribution of the values of <p{n)/n The local behavior of the Euler function 122 Problems on Chapter Chapter 9. Smooth Numbers Notation The smallest prime factor of an integer The largest prime factor of an integer The Rankin method An application to pseudoprimes The geometric method The best known estimates on W(x,y) 146

5 vi 9.8. The Dickman function Consecutive smooth numbers 149 Problems on Chapter Chapter 10. The Hardy-Ramanujan and Landau Theorems The Hardy-Ramanujan inequality Landau's theorem 159 Problems on Chapter Chapter 11. The abc Conjecture and Some of Its Applications The abc conjecture The relevance of the condition e > The Generalized Fermat Equation Consecutive powerful numbers Sums of fc-powerful numbers The Erdos-Woods conjecture A problem of Gandhi The k-abc conjecture 175 Problems on Chapter Chapter 12. Sieve Methods The sieve of Eratosthenes The Brun sieve Twin primes The Brun combinatorial sieve A Chebyshev type estimate The Brun-Titchmarsh theorem Twin primes revisited Smooth shifted primes The Goldbach conjecture The Schnirelman theorem The Selberg sieve The Brun-Titchmarsh theorem from the Selberg sieve The Large sieve Quasi-squares The smallest quadratic nonresidue modulo p 204 Problems on Chapter

6 I vii Chapter 13. Prime Numbers in Arithmetic Progression Quadratic residues The proof of the Quadratic Reciprocity Law Primes in arithmetic progressions with small moduli The Primitive Divisor theorem Comments on the Primitive Divisor theorem 227 Problems on Chapter Chapter 14. Characters and the Dirichlet Theorem Primitive roots Characters Theorems about characters L-series L(l,x) is finite if x is a non-principal character The nonvanishing of L(l, x): first step The completion of the proof of the Dirichlet theorem 244 Problems on Chapter Chapter 15. Selected Applications of Primes in Arithmetic Progression Known results about primes in arithmetical progressions Some Diophantine applications Primes p with p squarefree More applications of primes in arithmetic progressions Probabilistic applications 261 Problems on Chapter Chapter 16. The Index of Composition of an Integer Introduction Elementary results Mean values of A and 1/A Local behavior of A(n) Distribution function of A(n) Probabilistic results 276 Problems on Chapter Appendix: Basic Complex Analysis Theory Basic definitions 281

7 viii Infinite products The derivative of a function of a complex variable The integral of a function along a path The Cauchy theorem The Cauchy integral formula 289 Solutions to Even-Numbered Problems 291 Solutions to problems from Chapter Solutions to problems from Chapter Solutions to problems from Chapter Solutions to problems from Chapter Solutions to problems from Chapter Solutions to problems from Chapter Solutions to problems from Chapter Solutions to problems from Chapter Solutions to problems from Chapter Solutions to problems from Chapter Solutions to problems from Chapter Solutions to problems from Chapter Solutions to problems from Chapter Solutions to problems from Chapter Solutions to problems from Chapter Solutions to problems from Chapter Bibliography 405 Index 413