![]() Alternately, since every application of the method is so different, you might just try to understand it directly from the papers themselves.Ī good supplement to the last two books, and a good book in general, is Nathanson's Additive Number Theory, the Classical Bases which approaches some of the classical results of additive number theory, sometimes using the circle method. (This is another name for the circle method). Terence Tao won a Fields Medal in 2006 for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory. of an arithmetic function, as well as the associated natural density. Silverman lays out five basic steps in number theory: Accumulate mathematical or abstract data. This is pitched really high, and it's extremely likely that one would need to backtrack and fill in gaps while attempting to go through it.įor the circle method itself, I would recommend Vaughan's The Hardy-Littlewood Method. In elementary multiplicative number theory, which is the focus of this set of notes, particular emphasis is given on the following two statistics of a given arithmetic function : The summatory functions. Brown University mathematics professor Joseph H. ![]() The best book to understand Tao's papers is Tao and Vu's Additive Combinatorics. With these, you could understand Apostol's Introduction to Analytic Number Theory and then perhaps Montgomery and Vaughan's Multiplicative Number Theory.įor additive number theory, you want to understand algebra, real analysis, and probability theory. A Close Call: How a Near Failure Propelled Me to Succeed Terence Tao For as long as I can remember, I was always fascinated by numbers and the formal symbolic operations of mathe- matics, even before I knew the uses of mathematics in the real world. So if you want to read your way towards understanding the papers, this would be a good trio to go towards.įor analytic number theory, you want to understand some algebra and some complex analysis. He has received a number of awards, including the Salem Prize in. Tao's papers often evidence a good understanding and use of analytic number theory, additive number theory, and the circle method. Taos areas of research include harmonic analysis, PDE, combinatorics, and number theory. Terence Tao has made important contributions to a very wide range of mathematical fields, but has perhaps made the most impact in number theory.
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