# Ramanujan’s Contribution: A Technical Report

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M.L. Thivagar
2016, 244pp
Hardcover, 978938588302
Rs. 1695.00

This book provides an introduction to the large expanse of Ramanujan’s work in Number Theory, Analysis, and several other areas of Mathematics. The first chapter discusses about trigonometric sums and Ramanujan’s significant formulae. Ramanujan’s own proof of “P(5m + 4) is divisible by 5” is explained in chapter second. The Ramanujan differential equations in third chapter lead on to the Weierstrass invariants, and from there the final chapter, which provides a full account of elliptic functions as viewed by Ramanujan.

Needless to say, we only covered only a very small fraction of Ramanujan’s work. However, after developing a few facts about his work, we equipped to prove many interesting theorems. Our intent here is not to give a rigorous course in analysis but to emphasize the most important ideas about his work and how they interplay with other areas of Mathematics.

Dr. M.L. Thivagar, Professor and Chairperson of School of Mathematics, Madurai Kamaraj University, Madurai, India.

1 Ramanujan’s Contribution
On Ramanujan’s Trigonometrical Sums
Introduction
Generalizations of C(n,r)
Ramanujan’s Theorems
Identities
Arithmetic Functions
Introduction
Even Functions (Mod r)
A Linear Congruence
Ramanujan’s Significant Formula
Introduction
Relationships between H1(s) through H12(s)
Evaluation of H1(s) through H12(s) for Certain Values of s
The Case s = 2
The Case s = 3 and Ramanujan’s G(1)
Ramanujan’s Formula for ζ(2n+1)
A Proof of (1.3.6.4)
Proof of (1.3.6.1) for n≥1
Proof of (1.3.6.3)
Another Proof of (1.3.6.1) for n≤-1
Acknowledgement
Appendix A
A1 Partial Fraction Decomposition of Cotangent
A2 Ramanujan’s Formulae
A3 Bernoulli Numbers
A4 A Formula for ζ(1-2n)
A5 Halphen’s Formula
A6 Eisenstein Series

2 Ramanujan’s Theorem on p(5m+4)

Combinatorial Properties
Euler’s Identity
Product of Infinite Number of Power Series
Convergence Considerations
Jacobi’s Formula
Power Series with Integer Coefficients
Ramanujan’s Theorem

3 Ramanujan’s Differential Equations
The Basic Identity
The Differential Equations of P, Q, R
The Jordan – Kronecker Functions

4 Ramanujan’s Sense of Weierstrassian Invariants
Generalized Ramanujan’s Identity
Hyper-geometric Equations
Lambert Series Expansions

5 Ramanujan Sense of Development of Elliptic Functions
Elliptic Functions
Picard’s Theorem
Elliptic Integrals
Acknowledgement

Appendix B
B1
B2

Reference