He began publishing papers in 1941 when On Demlo numbers appeared in print. The problem he investigated in this paper was that of describing the digits of the product of two factors in terms of the digits of the factors. The paper examines some special cases. Further papers were Congruence properties of s(n), the sum of the divisors of n (1943), Multiplicative arithmetic functions (1943), Congruence peoperties of σa(n) (1945), Some applications of Ramanujan's trigonometrical sum Cm(n) (1944) and Congruence properties of Ramanujan's function t(n) (1945). Reviewing this last paper, Derrick Lehmer writes:-
This paper is concerned with a sum which is, in fact, the sum of the nth powers of the primitive mth roots of unity. The author points out its connection with partitions m, the sum in question being the excess of the number of partitions of n into an even number of incongruent parts modulo m over those into an odd number of such parts. Simple proofs are given of a number of known theorems, such as the one that asserts that the product of 2sin π( n/m) taken over all n less than and prime to m has the value p or 1 according as m is or is not a power of the prime p.The turning point in his career came when he went to the Institute for Advanced Study at Princeton. There he undertook further study and also was employed as an assistant to Hermann Weyl. At Princeton Ramanathan undertook research for a doctorate advised by Emil Artin. He was also heavily influenvced by Carl Siegel and much of the research he undertook in the years following his stay at Princeton was influenced by Siegel's mathematics. He submitted his doctoral thesis The Theory of Units of Quadratic and Hermitian Forms in 1951 and was awarded the degree for this work in which, among other things, he studied properties unit groups such as their finite generation and the finiteness of their covolume. After this he returned to India where he worked at the Tata Institute of Fundamental Research in Bombay from 1951 onwards. After a series of papers published in North American journals such as Identities and congruences of Ramanujan type (1950), The Theory of Units of Quadratic and Hermitian Forms (1951), Abelian quadratic forms (1952), and Units of quadratic forms (1952), he returned to publishing his research in Indian journals. For example he published A note on symplectic complements (1954), The Riemann sphere in matrix spaces (1955) and Quadratic forms over involutorial division algebras (1956) in the Journal of the Indian Mathematical Society.
Ramanathan built up a strong school of number theory at the Tata Institute of Fundamental Research. Raghavan writes :-
His abiding enthusiasm for the propagation of good mathematics and the spread of wholesome mathematical culture has been much instrumental in the moulding and flowering of several fine mathematicians and in the betterment of teaching and pursuit of research in Mathematics in many of our universities.The work of Ramanujan motivated a number of Ramanathan's early papers and later in his career he began to study his unpublished contributions :-
For several years, Professor Ramanathan had been actively interested in the study of published and unpublished work of Srinivasa Ramanujan, expounding, elucidating and extending Ramanujan's beautiful work on singular values of certain modular functions, Rogers-Ramanujan continued fractions and hypergeometric series. Many mathematicians in the West had made a tremendous advance in respect of many aspects of Ramanujan's unpublished work. In the light of the mathematical prospects so unveiled, he strongly urged many colleagues in India to take seriously to this fascinating domain, even if such activity might be cold-shouldered by "peers" from within.A biography of Ramanujan, as well as descriptions of some topics to illustrate Ramanujan's contributions, are given in Ramanujan's Notebooks (1987). An example of a paper motivated by his study of Ramanujan's work is Hypergeometric series and continued fractions (1987). R A Askey writes in a review:-
When trying to understand, or to obtain a continued fraction expansion, or when trying to evaluate a continued fraction, one natural method is to look at the three-term recurrence relation that generates the continued fraction and try to find hypergeometric functions whose contiguous relations contain the recurrence relation, or try to find three-term contiguous relations which are then used to generate continued fractions. This is done for some of Ramanujan's continued fractions ... Once this is done, there is almost surely a basic hypergeometric extension. Heine started the systematic search for such hypergeometric functions. The author extends much of the work mentioned above to basic hypergeometric series. The resulting continued fractions have many important special cases, including some found by Eisenstein, Heine, Rogers, Ramanujan, Selberg, Andrews and others.Another fascinating paper is Ramanujan's modular equations (1990). B D C Berndt writes:-
The present author commences with a very informative historical survey of modular equations. ... Of course, in a paper of only 18 pages in length, the author can only discuss a small portion of Ramanujan's modular equations and he concentrates therefore on equations of composite degree. He gives some proofs, shows connections to previous work, and offers insights into how Ramanujan may have discovered some of his equations. ...For most of his life his health had been smewhat poor and he developed serious illnesses in the last few years of his life following his retirement from the Tata Institute in December 1985. He underwent cerebral surgery and suffereed from Parkinson's disease.
Ramanathan received many honours. He was elected a Fellow of the Indian National Science Academy and of the Indian Academy of Sciences. In addition he was a Founder Fellow of the Maharashtra Academy of Sciences. He received a number of prizes including the Shanti Swarup Bhatnagar Prize, the Jawaharlal Nehru Fellowship, the Indian National Science Academy's Homi Bhabha Medal and the Padma Bhushan. He was a strong supported of the Indian Mathematical Society and served a term as its President. He was also Life President of the Bombay Mathematical Colloquium. He served the mathematical community in his role as editor of the Journal of the Indian Mathematical Society for more than 10 years, and in addition he was a member of the Editorial Board for Acta Arithmetica for around thirty years.
As to his character we quote from :-
His interests in English, Telugu and Tamil literature with his unfailing knack for pulling out apt quotations were just as remarkable as his erudition in music. A good conversationalist, he had been heard to remark a couple of times in his later years that the reason for his company being sought was probably that he was considered to be "well-rounded"! However, his occasional quips could have put off a few. He shunned publicity as much as he abhorred those who craved for power and ephemeral glory through the media; those who happened to know him somewhat closely could not have failed to note his simplicity and inner humility.We end this biography by quoting from the K G Ramanathan Memorial Issue of the Proceedings of the Indian Academy of Sciences which was published in February 1994:-
Professor K G Ramanathan was small of build but had a big influence on the post-independence Indian mathematical scene. Despite the legacy of the legendary Srinivasa Ramanujan and several other mathematicians of high standing early in this century, pursuit of mathematics had remained rather weak in Indian till the fifties. He was one of the few people responsible for the fortification which has put India firmly back on the international mathematical map. He not only was himself a front-ranking mathematician of international reputation, but also contributed a great deal to the emergence of a strong mathematical base at the Tata Institute of Fundamental Research as also to the overall development of research and teaching of mathematics in India and, to an extent, even beyond our shores. He was well recognized for his achievements in Number Theory, especially the analytic and arithmetic theory of quadratic forms over division algebras with involution.
Article by: J J O'Connor and E F Robertson
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