The Circle Method
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.
- 28.J.H. Bruinier, K. Ono, An algebraic formula for the partition function (in press) Google Scholar
- 29.J.H. Bruinier, K. Ono, Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms (in press) Google Scholar
- 42.M. Dewar, M.R. Murty, A derivation of the Hardy–Ramanujan formula from an arithmetic formula. Proc. Amer. Math. Soc. (in press) Google Scholar
- 52.L. Euler, Opera Postuma, vol. 1, (1862), pp. 203–204 Google Scholar
- 156.S.S. Pillai, Collected Papers, vol. 1, ed. by R. Balasubramanian, R. Thangadurai (Ramanujan Mathematical Society, India, 2010), p. xiii Google Scholar
- 181.A. Selberg, Reflections around the Ramanujan centenary, in Collected Papers, vol. 1 (Springer, Berlin, 1989) Google Scholar