Number Theory: Dreaming in Dreams: Proceedings of the 5th China-Japan Seminar, Higashi-Osaka, Japan, 27-31 August 2008 (Series on Number Theory and Its Applications)

This volume aims at collecting survey papers which give broad and enlightening perspectives of various aspects of number theory. Kitaoka's paper is a continuation of his earlier paper published in the last proceedings and pushes the research forward. Browning's paper introduces a new direction of research on analytic number theory - quantitative theory of some surfaces and Bruedern et al's paper details state-of-the-art affairs of additive number theory. There are two papers on modular forms - Kohnen's paper describes generalized modular forms (GMF) which has some applications in conformal field theory, while Liu's paper is very useful for readers who want to have a quick introduction to Maass forms and some analytic-number-theoretic problems related to them. Matsumoto et al's paper gives a very thorough survey on functional relations of root system zeta-functions, Hoshi-Miyake's paper is a continuation of Miyake's long and fruitful research on generic polynomials and gives rise to related Diophantine problems, and Jia's paper surveys some dynamical aspects of a special arithmetic function connected with the distribution of prime numbers. There are two papers of collections of problems by Shparlinski on exponential and character sums and Schinzel on polynomials which will serve as an aid for finding suitable research problems. Yamamura's paper is a complete bibliography on determinant expressions for a certain class number and will be useful to researchers. Thus the book gives a good-balance of classical and modern aspects in number theory and will be useful to researchers including enthusiastic graduate students. CONTENTS......Page 14 Preface......Page 7 1. Introduction......Page 16 2. Geometry of V0......Page 19 3. Overview of the proof......Page 21 3.1. Reduction to conics of low height......Page 22 3.2. Parametrisation of the conics......Page 24 3.3. Lattice point counting in the plane......Page 25 3.4. Divisor problem for binary forms......Page 26 3.5. Comparison with Peyre’s constant......Page 28 4. Further exploration......Page 31 References......Page 33 1.1. Diophantine inequalities......Page 35 1.2. Additive cubic forms......Page 36 1.3. Linear forms in primes......Page 40 1.4. Further applications......Page 42 1.5. A related diophantine inequality......Page 43 2.1. Some classical integrals......Page 44 2.2. Counting solutions of diophantine inequalities......Page 45 2.3. Weighted counting......Page 47 2.4. The central interval......Page 50 2.5. The interference principle......Page 53 3.1. Plancherel’s identity......Page 55 3.2. Some mean values......Page 56 3.3. The amplification technique......Page 58 3.4. Linear forms in primes......Page 59 3.5. Bessel’s inequality......Page 60 4.2. Exponential sums over test sequences......Page 61 4.3. Potential applications......Page 63 5.1. An illustrative example......Page 64 5.2. A quadratic average......Page 66 5.4. An inequality involving quadratic polynomials......Page 69 5.5. An application of Vinogradov’s method......Page 70 5.6. Linear forms in primes, yet again......Page 71 6.1. Smooth cubic Weyl sums......Page 73 6.2. Senary cubic forms......Page 74 6.3. Two technical estimates......Page 76 6.4. The lower bound variant......Page 77 6.5. An auxiliary inequality......Page 80 6.6. Additive forms of large degree......Page 83 6.7. Proof of Theorem 1.8......Page 85 6.8. Proof of Theorem 1.9......Page 86 7.1. The counting integral......Page 89 7.3. The complementary compositum......Page 90 References......Page 91 1. Introduction......Page 95 2. Dynamics of the w function......Page 96 3. Inverse problem......Page 97 4. The sketch of the proof of Theorem 3.5......Page 99 References......Page 101 Some Diophantine problems arising from the isomorphism problem of generic polynomials Akinari Hoshi and Katsuya Miyake......Page 102 1. Introduction......Page 103 2. Some results: the cubic case......Page 105 3. A parametric family of Thue equations......Page 107 4. The case of D4......Page 110 5. Numerical examples: the case of C4......Page 111 6. Numerical examples: the case of D5......Page 112 7. Appendix......Page 116 References......Page 118 1. Introduction and Conjectures......Page 121 1.1. Irreducible case......Page 127 1.2. Reducible case......Page 131 2.1. n = 3......Page 132 2.4. n = 6......Page 133 2.6. n = 8......Page 134 2.7. n = 9......Page 135 2.9. n = 12......Page 136 2.10. n = 15......Page 139 2.11. Other examples......Page 140 References......Page 141 2. Generalized modular functions, main features of the theory and examples......Page 142 3. Fourier coefficients of GMF’s......Page 146 References......Page 149 2. A method to evaluate the Riemann zeta-function......Page 150 3. Functional relations for ζ2(s1, s2, s3;A2)......Page 157 4. Another method to construct functional relations for Dirichlet series......Page 161 5. A general form of functional relations......Page 163 6. Some lemmas for explicit construction of functional relations......Page 166 7. Functional relations for ζ3(s;A3)......Page 170 8. Functional relations for ζ2(s;C2)......Page 179 9. Functional relations for ζ3(s;B3) and for ζ3(s;C3)......Page 183 References......Page 195 1.1. The aim of the paper......Page 199 2. Maass forms for SL2(Z)......Page 200 3.1. Fourier expansion of Maass forms......Page 203 3.2. Analytic continuation of Eisenstein series......Page 205 4. Spectral decomposition of non-Euclidean Laplacian......Page 209 5. Hecke’s theory for Maass forms......Page 214 6. The Kuznetsov trace formula......Page 216 7.1. Automorphic L-functions attached to Maass forms......Page 218 8. Maass forms for Γ0(N) and their L-functions......Page 220 8.1. Maass forms for Γ0(N)......Page 221 8.2. Automorphic L-functions for Maass forms for Γ0(N)......Page 222 9.2. A Linnik-type problem for Maass forms......Page 224 9.3. A Linnik-type problem for holomorphic forms......Page 225 9.4. Linnik-type problems for higher rank groups......Page 226 Acknowledgements.......Page 229 References......Page 230 The number of non-zero coefficients of a polynomial-solved and unsolved problems Andrzej Schinzel......Page 232 References......Page 236 1. Introduction......Page 237 2. Notation......Page 238 3.1. Exponential functions......Page 239 3.2. Short character sums......Page 241 3.3. Smooth numbers, S-units and primes......Page 243 3.4. Combinatorial sequences......Page 244 3.5. Polynomial discriminants......Page 246 3.6. Arithmetic functions......Page 247 3.7. Beatty sequences......Page 248 3.8. Sparse polynomials......Page 249 3.9. Nonlinear recurrence sequences......Page 251 Acknowledgements......Page 252 References......Page 253 Errata to “A general modular relation in analytic number theory” Haruo Tsukada......Page 258 References......Page 259 Author Index......Page 266 This volume aims at collecting survey papers which give broad and enlightening perspectives of various aspects of number theory. Kitaoka's paper is a continuation of his earlier paper published in the last proceedings and pushes the research forward. Browning's paper introduces a new direction of research on analytic number theory - quantitative theory of some surfaces and Bruedern et al 's paper details state-of-the-art affairs of additive number theory. There are two papers on modular forms - Kohnen's paper describes generalized modular forms (GMF) which has some applications in conformal fi 7. An appendix on inhomogenous polynomials7.1. The counting integral; 7.2. The central interval; 7.3. The complementary compositum; Acknowledgements; References; Recent progress on dynamics of a special arithmetic function Chaohua Jia; 1. Introduction; 2. Dynamics of the w function; 3. Inverse problem; 4. The sketch of the proof of Theorem 3.5; 5. Acknowledgements; References; Some Diophantine problems arising from the isomorphism problem of generic polynomials Akinari Hoshi and Katsuya Miyake; 1. Introduction; 2. Some results: the cubic case; 3. A parametric family of Thue equations 1.1. Diophantine inequalities1.2. Additive cubic forms; 1.3. Linear forms in primes; 1.4. Further applications; 1.5. A related diophantine inequality; 2. The Fourier transform method; 2.1. Some classical integrals; 2.2. Counting solutions of diophantine inequalities; 2.3. Weighted counting; 2.4. The central interval; 2.5. The interference principle; 3. Classical mean square methods; 3.1. Plancherel's identity; 3.2. Some mean values; 3.3. The amplification technique; 3.4. Linear forms in primes; 3.5. Bessel's inequality; 4. Semi-classical averaging; 4.1. Another mean square approach 4.2. Exponential sums over test sequences4.3. Potential applications; 5. Fourier analysis of exceptional sets; 5.1. An illustrative example; 5.2. A quadratic average; 5.3. Some brief heckling; 5.4. An inequality involving quadratic polynomials; 5.5. An application of Vinogradov's method; 5.6. Linear forms in primes, yet again; 6. Outstanding arts; 6.1. Smooth cubic Weyl sums; 6.2. Senary cubic forms; 6.3. Two technical estimates; 6.4. The lower bound variant; 6.5. An auxiliary inequality; 6.6. Additive forms of large degree; 6.7. Proof of Theorem 1.8; 6.8. Proof of Theorem 1.9 4. The case of D45. Numerical examples: the case of C4; 6. Numerical examples: the case of D5; 7. Appendix; References; A statistical relation of roots of a polynomial in different local fields II Yoshiyuki Kitaoka; 1. Introduction and Conjectures; 1.1. Irreducible case; 1.2. Reducible case; 1.3. Generalization; 2. Numerical data; 2.1. n = 3; 2.3. n = 5; 2.4. n = 6; 2.5. n = 7; 2.6. n = 8; 2.7. n = 9; 2.8. n = 10; 2.9. n = 12; 2.10. n = 15; 2.11. Other examples; References; Generalized modular functions and their Fourier coefficients Winfried Kohnen; 1. Introduction Preface; CONTENTS; Recent progress on the quantitative arithmetic of del Pezzo surfaces Tim D. Browning; 1. Introduction; Acknowledgements; 2. Geometry of V0; 3. Overview of the proof; 3.1. Reduction to conics of low height; 3.2. Parametrisation of the conics; 3.3. Lattice point counting in the plane; 3.4. Divisor problem for binary forms; 3.5. Comparison with Peyre's constant; 4. Further exploration; References; Additive representation in thin sequences, VIII: Diophantine inequalities in review Jorg Brudern, Koichi Kawada and Trevor D. Wooley; 1. Theme and results Aims at collecting survey papers which give broad perspectives of various aspects of number theory. This work introduces a fresh direction of research on analytic number theory - quantitative theory of some surfaces. It describes generalized modular forms (GMF) which has some applications in conformal field theory. Editors Takashi Aoki, Shigeru Kanemitsu, Jianya Liu. Includes Bibliographical References And Index.