By J. W. S. Cassels

This tract units out to provide a few proposal of the elemental innovations and of a few of the main impressive result of Diophantine approximation. a range of theorems with whole proofs are offered, and Cassels additionally offers an actual advent to every bankruptcy, and appendices detailing what's wanted from the geometry of numbers and linear algebra. a few chapters require wisdom of components of Lebesgue conception and algebraic quantity thought. it is a precious and concise textual content geared toward the final-year undergraduate and first-year graduate scholar

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2). 1; see [3, p. 1] for a proof. 3. 2). 1 on its interior. If s is any complex number and Re a, Re b > 0, then ∞ 2 b λn + a a(n) n=1 ∞ = s/2 K s 2 (λn + a)b x s−1 e−ax−b/x P(x)d x 0 ∞ +2 b(n) n=1 a μn + b (r −s)/2 K r −s 2 (μn + b)a . 2 Formulas of Koshliakov and Guinand 39 Example 1. Let rk (n) denote the number of representations of the positive integer n as a sum of k squares, where k ≥ 2. Then ∞ ϕ(s) := ζk (s) = rk (n)n −s , σ > 1, n=1 denotes the Epstein zeta function associated with the quadratic form n 21 +n 22 +· · ·+n 2k .

C. Berndt, Ramanujan’s Notebooks, Part II, Springer–Verlag, New York, 1989. 7. C. Berndt, Ramanujan’s Notebooks, Part III, Springer–Verlag, New York, 1991. 8. C. Berndt, An unpublished manuscript of Ramanujan on infinite series identities, J. Ramanujan Math. Soc. 19 (2004), 57–74. 9. C. J. Yee, Ramanujan’s contributions to Eisenstein series, especially in his lost notebook, in Number-Theoretic Methods – Future Trends, C. Jia and S. , Kluwer, Dordrecht, 2002, pp. 31–53; abridged version, A survey on Eisenstein series in Ramanujan’s lost notebook, in New Aspects of Analytic Number Theory, Y.

In complex analysis the main utility is in the theory of compact Riemann surfaces, with which we do not deal. We do show how identities among theta functions yield proofs of Picard’s theorem and a conformal map of the rectangle onto the disk. Key words: Theta functions, triangular numbers, partitions, Riemann map, Picard. 2000 Mathematics Subject Classifications: 11A, 11A25, 11B65, 30C 1 Introduction This chapter is a slight elaboration of the lectures given by the author in Gainesville at the special year in 2005.