Goldbach’s Problem

1. Front Matter

   Pages i-xv

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2. Introduction

   • Michael Th. Rassias

   Pages 1-5

3. Step-by-Step Proof of Vinogradov’s Theorem

   • Michael Th. Rassias

   Pages 7-65

4. The Ternary Goldbach Problem with a Prime and Two Isolated Primes

   • Michael Th. Rassias

   Pages 67-86

5. Basic Steps of the Proof of Schnirelmann’s Theorem

   • Michael Th. Rassias

   Pages 87-97

6. Back Matter

   Pages 99-122

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Important results surrounding the proof of Goldbach's ternary conjecture are presented in this book. Beginning with an historical perspective along with an overview of essential lemmas and theorems, this monograph moves on to a detailed proof of Vinogradov's theorem. The principles of the Hardy-Littlewood circle method are outlined and applied to Goldbach's ternary conjecture. New results due to H. Maier and the author on Vinogradov's theorem are proved under the assumption of the Riemann hypothesis. The final chapter discusses an approach to Goldbach's conjecture through theorems by L. G. Schnirelmann. This book concludes with an Appendix featuring a sketch of H. Helfgott's proof of Goldbach's ternary conjecture. The Appendix also presents some biographical remarks of mathematicians whose research has played a seminal role on the Goldbach ternary problem.

 The author's step-by-step approach makes this book accessible to those that have mastered classical number theory and fundamental notions of mathematical analysis. This book will be particularly useful to graduate students and mathematicians in analytic number theory, approximation theory as well as to researchers working on Goldbach's problem.

• Goldbach’s conjecture
• Hardy-Littlewood circle method
• Prime Number Theorem
• Vaughan’s proof
• Vinogradov’s theorem
• Ternary Goldbach Conjecture
• TGC
• Binary Goldbach Conjecture
• BGC
• The Circle Method
• Schnirelmann's theorem

“The author masterfully introduces the uninitiated reader to this beautiful but also demanding subject of mathematics in a step-by-step, self-contained manner. Throughout the book, the reader is also lead to explore more advanced aspects of the problems studied. The topics treated, the structure and the presentation of this book are such that it would be excellent for class as well as seminar use.” (Vijay Gupta, European Mathematical Society, euro-math-soc.eu, Issue 115, March, 2020)

“The monograph is well readable since the author provides many details and thus makes the presentation self-contained. The book will be a valuable addition to every mathematical library. It will serve as an invaluable source to students and professors interested in the subject. It is an important contribution to the mathematical literature.” (Helmut Maier, zbMATH, 2017)

• Department of Mathematics, ETH Zürich, Zürich, Switzerland

  Michael Th. Rassias

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