Robert A. Rankin

Ramanujan : letters and commentary

A brief biography of Ramanujan Ramanujan in Madras (chapter 1) Ramanujans first two letters to Hardy and Hardys response (chapter 2) Preparing to go to England (chapter 3) Ramanujan at Cambridge (chapter 4) Ramanujan is ill (chapter 5) Ramanujan returns to India (chapter 6) After his death (chapter 7) Ramanujans papers and manuscripts (chapter 8) Family histories (chapter 9) Provenance of letters References Index.

The Man Who Knew Infinity

On a flight to Kansas this week, where I was participating in a scholarly communications symposium, I watched The Man Who Knew Infinity [2016]; a dramatisation of the life of the Indian mathematician Srinivasa Ramanujan as he travels to Cambridge and is eventually made a fellow of the Royal Society. I had previously encountered Ramanujans extraordinary story when I had seen A Disappearing Number, a devised dramatic piece by Théâtre de Complicité and it has stuck with me ever since.

A Minimum Problem for the Epstein Zeta-Function

In some recent work by D. G. Kendall and the author † on the number of points of a lattice which lie in a random circle the mean value of the variance emerged as a constant multiple of the value of the Epstein zeta-function Z(s) associated with the lattice, taken at the point s= . Because of the connexion with the problems of closest packing and covering it seemed likely that the minimum value of Z ( ) would be attained for the hexagonal lattice; it is the purpose of this paper to prove this and to extend the result to other real values of the variable s .

Diagonalizing Eisenstein Series. I

In this paper we consider the action of Hecke operators T n (n ∈ IN), and their adjoint operators T* n , on Eisenstein series belonging to the group Γ0(N) and having integral weight k > 2 and arbitrary character χ modulo N. It is shown that the space ɛ k (x) spanned by these Eisenstein series splits up into a number of subspaces ɛ k (x,t)> where t is a divisor of N, each being invariant under the operators T n and T* n with (n, N) = 1. If x is a primitive character modulo N, this holds also for T n with (n, N) > 1, but this need not be true for general x modulo N. A basis of modular forms that are eigenfunctions for T n with (n, N) = 1 is constructed for each appropriate t and explicit evaluations of G L \T n are given for each Eisenstein series G L (L ∈ Γ(l)) and any positive integer n prime to N, or any n that is a prime divisor of N, the results being particularly simple when N is squarefree. The corresponding results for G L \T* n when (n, N) > 1 will be given in a subsequent paper.

CONVERGENCE AND UNIFORMITY

A subsequence of a convergent sequence converges to the same limit. On the other hand, the convergence of a given subsequence does not imply that the original sequence converges. Every bounded real sequence possesses a convergent subsequence. Every bounded complex sequence possesses a convergent subsequence. The convergence tests that have been obtained are applicable only to series of non-negative terms or to test for absolute convergence. A test has been given for convergence, which is more powerful than the tests of Cauchy and D’Alembert and applies to series whose terms are positive and of a particularly regular form. This chapter discusses some applications of uniform convergence to improper integrals.

Sums of Squares: An Elementary Method

If x 1, x 2, ... , x s are integers positive negative or zero such that

LIMITS AND CONTINUITY

x_1^2 + x_2^2 + \cdot \cdot \cdot x_s^2 = n,

CHAPTER 3 – DIFFERENTIABILITY

then (x 1, x 2, ... , x s ) is called a representation of n as a sum of s squares, and the total number of representations is denoted by R s (n). Two representations (x 1, x 2, ... , x s ) and (y 1, y 2, ... , y s ) are considered to be different unless