Ebook: Number Theoretic Methods: Future Trends

This volume contains the proceedings of the very successful second China-Japan Seminar held in lizuka, Fukuoka, Japan, during March 12-16, 2001 under the support of the Japan Society for the Promotion of Science (JSPS) and the National Science Foundation of China (NSFC), and some invited papers of eminent number-theorists who visited Japan during 1999-2001 at the occasion of the Conference at the Research Institute of Mathematical Sciences (RIMS), Kyoto University. The proceedings of the 1st China-Japan Seminar held in September 1999 in Beijing has been published recently {2002) by Kluwer as DEVM 6 which also contains some invited papers. The topics of that volume are, however, restricted to analytic number theory and many papers in this field are assembled. In this volume, we return to the lines of the previous one "Number Theory and its Applications", published as DEVM 2 by Kluwer in 1999 and uphold the spirit of presenting various topics in number theory and related areas with possible applica­ tions, in a unified manner, and this time in nearly a book form with a well-prepared index. We accomplish this task by collecting highly informative and readable survey papers (including half-survey type papers), giving overlooking surveys of the hith­ erto obtained results in up-to-the-hour form with insight into the new developments, which are then analytically continued to a collection of high standard research papers which are concerned with rather diversed areas and will give good insight into new researches in the new century.

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This book contains various topics in number theory and provides the reader with an overvierw of current and future researches in the field.
Audience: Researchers and graduate students in number theory, enthusiastic amateurs and undergraduate students who have some basic knowledge in mathematics.

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This book contains various topics in number theory and provides the reader with an overvierw of current and future researches in the field.
Audience: Researchers and graduate students in number theory, enthusiastic amateurs and undergraduate students who have some basic knowledge in mathematics.
Content:
Front Matter....Pages i-xi
A Limiting form of the q-Dixon 4?3 Summation and Related Partition Identities....Pages 1-14
Arithmetical Properties of Solutions of Linear Second Order q-Difference Equations....Pages 15-29
Ramanujan’s Contributions to Eisenstein Series, Especially in His Lost Notebook....Pages 31-53
New Applications of a Result of Galochkin on Linear Independence....Pages 55-66
Partitions Modulo Prime Powers and Binomial Coefficients....Pages 67-72
Infinite Sums, Diophantine Equations and Fermat’s Last Theorem....Pages 73-95
On the Nature of the “Explicit Formulas” in Analytic Number Theory — A Simple Example....Pages 97-118
Product Representations by Rationals....Pages 119-150
On the Distribution of ?p Modulo One....Pages 151-157
Ramanujan’s Formula and Modular Forms....Pages 159-212
Waldspurger’s Formula and Central Critical Values of L-Functions of Newforms in Weight Aspect....Pages 213-217
Primitive Roots: A Survey....Pages 219-231
Zeta-Functions Defined by Two Polynomials....Pages 233-262
Some Aspects of Interactions between Algebraic Number Theory and Analytic Number Theory....Pages 263-299
On G-Functions and Pad? Approximations....Pages 301-310
A Penultimate Step toward Cubic Theta-Weyl Sums....Pages 311-338
Some Results in View of Nevanlinna Theory....Pages 339-350
A Historical Comment about the GVT in Short Interval....Pages 351-367
Convexity and Intersection of Random Spaces....Pages 369-380
Generalized Hypergeometric Series and the Symmetries of 3-j and 6-j Coefficients....Pages 381-403
Stability and New Non-Abelian Zeta Functions....Pages 405-419
A Hybrid Mean Value of L-Functions and General Quadratic Gauss Sums....Pages 421-434
Back Matter....Pages 435-441

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This book contains various topics in number theory and provides the reader with an overvierw of current and future researches in the field.
Audience: Researchers and graduate students in number theory, enthusiastic amateurs and undergraduate students who have some basic knowledge in mathematics.
Content:
Front Matter....Pages i-xi
A Limiting form of the q-Dixon 4?3 Summation and Related Partition Identities....Pages 1-14
Arithmetical Properties of Solutions of Linear Second Order q-Difference Equations....Pages 15-29
Ramanujan’s Contributions to Eisenstein Series, Especially in His Lost Notebook....Pages 31-53
New Applications of a Result of Galochkin on Linear Independence....Pages 55-66
Partitions Modulo Prime Powers and Binomial Coefficients....Pages 67-72
Infinite Sums, Diophantine Equations and Fermat’s Last Theorem....Pages 73-95
On the Nature of the “Explicit Formulas” in Analytic Number Theory — A Simple Example....Pages 97-118
Product Representations by Rationals....Pages 119-150
On the Distribution of ?p Modulo One....Pages 151-157
Ramanujan’s Formula and Modular Forms....Pages 159-212
Waldspurger’s Formula and Central Critical Values of L-Functions of Newforms in Weight Aspect....Pages 213-217
Primitive Roots: A Survey....Pages 219-231
Zeta-Functions Defined by Two Polynomials....Pages 233-262
Some Aspects of Interactions between Algebraic Number Theory and Analytic Number Theory....Pages 263-299
On G-Functions and Pad? Approximations....Pages 301-310
A Penultimate Step toward Cubic Theta-Weyl Sums....Pages 311-338
Some Results in View of Nevanlinna Theory....Pages 339-350
A Historical Comment about the GVT in Short Interval....Pages 351-367
Convexity and Intersection of Random Spaces....Pages 369-380
Generalized Hypergeometric Series and the Symmetries of 3-j and 6-j Coefficients....Pages 381-403
Stability and New Non-Abelian Zeta Functions....Pages 405-419
A Hybrid Mean Value of L-Functions and General Quadratic Gauss Sums....Pages 421-434
Back Matter....Pages 435-441
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