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The circle method, pioneered by Ramanujan and Hardy in the early 20th century, has over the past 100 years become part of the standard tool chest of analytic number theory. Its scope of applications is ever-expanding, and the subject continues to see important breakthroughs. This book provides an introduction to the circle method that is accessible to undergraduate students with no background in number theory. The authors' goal is to show the students the elegance of the circle method and at the same time give a complete solution of the famous Waring problem as an illustration of the method. The first half of this book is a curated introduction to elementary number theory with an emphasis on topics needed for the second half. The second half showcases the two most ""classic"" applications of the circle method, to Waring's problem (following Hardy-Littlewood-Hua) and to Goldbach's conjectures (following Vinogradov, with improvements by Vaughan). This text is suitable for a one-semester undergraduate course or for independent study and will be a great entry point into this fascinating area of research.
- Nyhet
Some Contributions to Number Theory and Beyond
Proceedings of the Centenary Symposium for M. V. Subbarao
Inbunden, Engelska, 2026
3 765 kr
Kommande
A symposium on number theory was organized at the Indian Institute of Science Education and Research Pune from July 12 - 16, 2021 in honour of Professor M. V. Subbarao (May 4, 1921 – February 15, 2006) on his birth centenary. Due to travel restrictions imposed by the COVID-19 pandemic, this symposium was conducted online and had participation from speakers all across the globe. The editors of this volume are glad to present the Proceedings of this symposium to readers.It contains twelve chapters contributed by symposium speakers, and a personal reminiscence of Professor Subbarao.These articles encompass a wide variety of topics, including those to which Prof. Subbarao has made fundamental contributions, such as partition theory, and the characterization of additive and multiplicative functions. Chapters touch upon wider themes in number theory, such as summation formulas arising from a boundary value problem in heat conduction, parity results for the generalized divisor function, probabilistic limit theorems for the zeta function, infinite-rank Euclidean lattices, the density of visible lattice points, and a question of Baker in transcendental number theory. Finally, in resonance with one of the most challenging problems that the world has faced (and continues to face) in recent times, this volume also contains an article on modelling pandemics with applications to COVID-19.