The Circle Method

M. Ram Murty and V. Kumar Murty
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M. Ram Murty: Queen’s University, Department of Mathematics and Statistics
V. Kumar Murty: University of Toronto, Department of Mathematics

Chapter Chapter 5 in The Mathematical Legacy of Srinivasa Ramanujan, 2013, pp 67-96 from Springer

Abstract: Abstract In his 1913 letter to G.H. Hardy, Ramanujan gave an asymptotic formula for the nth coefficient of the inverse of a theta function. This was the earliest result obtained using the circle method, the rudiments of which Ramanujan seemed to have had before his journey to England. In a famous paper of 1918, Hardy and Ramanujan developed this method to derive an asymptotic formula for the partition function. After Ramanujan’s untimely death, Hardy developed the method along with Littlewood and applied it to tackle other major unsolved problems, such as Waring’s problem and Goldbach’s problem. We discuss these problems in the context of the circle method.

Keywords: Circle Method; Schnirelmann Density; Ternary Goldbach Problem; Major Arcs; Generalized Riemann Hypothesis (GRH) (search for similar items in EconPapers)
Date: 2013
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-81-322-0770-2_5

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DOI: 10.1007/978-81-322-0770-2_5

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