Arithmetic Functions and Integer Products

This book has grown out of a course of lectures on elliptic functions, given in German, at the Swiss Federal Institute of Technology, Zurich, during the summer semester of 1982. Its aim is to give some idea of the theory of elliptic functions, and of its close connexion with theta-functions and modular functions, and to show how it provides an analytic approach to the solution of some classical problems in the theory of numbers. It comprises eleven chapters. The first seven are function-theoretic, and the next four concern arithmetical applications. There are Notes at the end of every chapter, which contain references to the literature, comments on the text, and on the ramifications, old and new, of the problems dealt with, some of them extending into cognate fields. The treatment is self-contained, and makes no special demand on the reader's knowledge beyond the elements of complex analysis in one variable, and of group theory. I. Periods Of Meromorphic Functions -- Ii. General Properties Of Elliptic Functions -- Iii. Weierstrass's Elliptic Function P(z) -- Iv. The Zeta-function And The Sigma-function Of Weierstrass -- V. The Theta-functions -- X. Table Of Contents -- Vi. The Modular Function J(.) -- Vii. The Jacobian Elliptic Functions And The Modular Function A(.) -- Viii. Dedekind's 1]-function And Euler's Theorem On Pentagonal Numbers -- Ix. The Law Of Quadratic Reciprocity -- X. The Representation Of A Number As A Sum Of Four Squares -- Xi. The Representation Of A Number By A Quadratic Form -- I. Periods Of Meromorphic Functions -- 1. Meromorphic Functions -- 2. Periodic Meromorphic Functions -- 3. Jacobi's Lemma -- 4. Elliptic Functions -- 5. The Modular Group And Modular Functions -- Ii. General Properties Of Elliptic Functions -- 1. The Period Parallelogram -- 2. Elementary Properties Of Elliptic Functions -- Iii. Weierstrass's Elliptic Function P(z) -- 1. The Convergence Of A Double Series -- 2. The Elliptic Function P(z) -- 3. The Differential Equation Associated With P(z) -- 4. The Addition-theorem -- 5. The Generation Of Elliptic Functions -- Appendix I. The Cubic Equation : -- Appendix Ii. The Biquadratic Equation : -- Iv. The Zeta-function And The Sigma-function Of Weierstrass -- 1. The Function ((z) -- 2. The Function A(z) -- 3. An Expression For Elliptic Functions -- V. The Theta-functions -- 1. The Function O(v, R) -- 2. The Four Sigma-functions -- 3. The Four Theta-functions -- 4. The Differential Equation -- X. Table Of Contents -- 5. Jacobi's Formula For ()f (0, .) -- 6. The Infinite Products For The Theta-functions -- 7. Theta-functions As Solutions Of Functional Equations -- 8. The Transformation Formula Connecting ()iv, .) And ()3(v, -f) -- Vi. The Modular Function J(.) -- 1. Definition Of J(.) -- 2. The Functions G2(.) And Gi.) -- 3. Expansion Of The Function J(.) And The Connexion With Theta-functions -- 4. The Function J(.) In A Fundamental Domain Of The Modular Group -- 5. Relations Between The Periods And The Invariants Of 8o(u) -- 6. Elliptic Integrals Of The First Kind -- Vii. The Jacobian Elliptic Functions And The Modular Function A(.) -- 1. The Functions Sn U, En U, Dn U Of Jacobi -- 2. Definition By Theta-functions -- 3. Connexion With The Sigma-functions -- 4. The Differential Equation -- 5. Infinite Products For The Jacobian Elliptic Functions -- 6. Addition-theorems For Sn U, En U, Dn U -- 7. The Modular Function A(.) -- 8. Mapping Properties Of A(.) And Picard's Theorem -- Viii. Dedekind's 1]-function And Euler's Theorem On Pentagonal Numbers -- 1. Connexion With The Invariants Of The So-function And With The Theta-functions -- 2. Euler's Theorem And Jacobi's Proof -- 3. The Transformation Formula Connecting 1](z) And 1]( -f) -- 4. Siegel's Proof Of Theorem 1 -- 5. Connexion Between 1](z) And The Modular Functions J(z), A(z) -- Ix. The Law Of Quadratic Reciprocity -- 1. Reciprocity Of Generalized Gaussian Sums -- 2. Quadratic Residues -- 3. The Law Of Quadratic Reciprocity -- X. The Representation Of A Number As A Sum Of Four Squares -- 1. The Theorems Of Lagrange And Of Jacobi -- 2. Proof Of Jacobi's Theorem By Means Of Theta-functions -- 3. Siegel's Proof Of Jacobi's Theorem -- Xi. The Representation Of A Number By A Quadratic Form -- 1. Positive-definite Quadratic Forms -- 2. Multiple Theta-series And Quadratic Forms -- 3. Theta-functions Associated To Positive-definite Forms -- 4. Representation Of An Even Integer By A Positive-definite Form. K. Chandrasekharan. Based On Lectures Given At The Swiss Federal Institute Of Technology, Zürich, During The Summer Semester Of 1982. Includes Bibliographies And Index. Cover......Page 1 Title Page......Page 4 Copyright Page......Page 5 Preface......Page 6 Acknowledgments......Page 8 Contents......Page 10 Notation......Page 14 Introduction......Page 18 Duality and the Differences of Additive Functions......Page 24 First Motive......Page 36 Multiplicative Functions......Page 40 Generalized Turan-Kubilius Inequalities......Page 44 Selberg's Sieve Method......Page 50 Kloosterman Sums......Page 51 CHAPTER 2 A Diophantine Equation......Page 54 CHAPTER 3 A First Upper Bound......Page 70 The First Inductive Proof......Page 81 The Second Inductive Proof......Page 88 Concluding Remarks......Page 93 CHAPTER 4 Intermezzo: The Group Q*/r......Page 95 Duality in Finite Spaces......Page 98 Self-adjoint Maps......Page 100 Duality in Hilbert Space......Page 109 Duality in General......Page 110 Second Motive......Page 114 The Large Sieve and Prime Number Sums......Page 118 The Method of Vinogradov in Vaughan's Form......Page 131 Dirichlet L-Series......Page 136 Additive Functions on Arithmetic Progressions......Page 138 Algebraicanalytic Inequalities......Page 166 CHAPTER 8 The Loop......Page 172 Third Motive......Page 194 CHAPTER 9 The Approximate Functional Equation......Page 200 The Basic Inequality......Page 221 The Decomposition of the Mean......Page 249 Concluding Remarks......Page 256 CHAPTER 11 Some Historical Remarks......Page 261 CHAPTER 12 From L2 to L-......Page 267 CHAPTER 13 A Problem of Katai......Page 276 CHAPTER 14 Inequalities in L-......Page 281 More Duality; Additive Functions as Characters......Page 294 Divisible Groups and Modules......Page 295 Sets of Uniqueness......Page 298 Algorithms......Page 304 CHAPTER 16 The Second Intermezzo......Page 308 A Ring of Operators......Page 314 Practical Measures......Page 324 CHAPTER 18 Simultaneous Product Representations by Values of Rational Functions......Page 326 Linear Recurrences in Modules......Page 327 Elliptic Power Sums......Page 335 Concluding Remarks......Page 345 CHAPTER 19 Simultaneous Product Representations with a;x+b;......Page 346 CHAPTER 20 Information and Arithmetic......Page 360 Transition to Arithmetic......Page 363 Information as an Algebraic Object......Page 370 CHAPTER 21 Central Limit Theorem for Differences......Page 373 CHAPTER 22 Density Theorems......Page 389 Groups of Bounded Order......Page 396 Measures on Dual Groups......Page 397 Arithmic Groups......Page 409 Concluding Remarks......Page 410 Exercises......Page 411 Unsolved Problems......Page 434 Progress in Probabilistic Number Theory......Page 440 Analogues of the Turan-Kubilius Inequality......Page 452 References......Page 466 Subject Index......Page 476