During his stay at Cambridge, Ananda Rau was first elected as an exhibitioner and then a scholar of King's College. His mathematical talent blossomed under the influence of G H Hardy who was then at Trinity College. Hardy wrote on 26 March 1918, "Mr K A Rau has spent the last year in mathematical research, undertaken in part under my advice. Some of his results are contained in an essay which obtained the Smith's Prize this year; and he has done other valuable work which will no doubt be published in due course."

The contents of this paper form a slightly modified version of part of an essay for which a Smith's Prize was awarded in 1918 in the University of Cambridge.

1. In a paper published several years ago in these Proceedings Hardy and Littlewood proved a series of theorems of great generality, whose nature they have called Tauberian; and in the same paper they applied these theorems to the study of a variety of problems. The theorems were concerned, to put the matter roughly, with the relations connecting the behaviour of a function and of its derivatives of integral orders. Hardy and Littlewood foresaw that theorems similar to theirs can be proved involving Riemann-Liouville derivatives of non-integral orders. M Riesz has since proved some of these extensions by making use of an important mean value theorem.

2. Among the applications which Hardy and Littlewood made of their Tauberian theorems, there is one class with which the present paper is connected, namely the applications to questions of convergence and summability of Dirichlet's series. Hardy and Littlewood concerned themselves only with ordinary Dirichlet's series, that is, series of the type $\sum a_{n} n^{-s}$; the methods of summation which they employed were those associated with the name of Cesàro, and the orders of summation were integral. But they foresaw not only that their theorems can be extended to non-integral orders of Cesàro summation, but also that the more general methods of summation by typical means introduced by Riesz can be employed and theorems of a similar character proved for Dirichlet's series of arbitrary type. Riesz, in the paper referred to, has proved some extensions of this nature.

3. The present paper deals with some of the generalisations suggested by Hardy and Littlewood and also some additional matter. The mean value theorem of Riesz and his extended forms of the Tauberian theorems of Hardy and Littlewood play an important part in the present paper.

The problem is then very much more difficult; Mr Hardy has kindly pointed out to me its close connection with the Prime Number Theory. He also tells me that he and Mr Littlewood have proved (on assuming the Prime Number Theorem) that, if a series is summable (L), then it is also summable (A). I understand that their paper on the subject is to be published in this volume of the Proceedings.

This theorem was previously known to Messrs Hardy and Littlewood, and I am indebted to them for their permission to have it published here.

Both often met at Cambridge. They talked to each other in homely Tamil. Ananda Rau never ceased to regret that he was then not mature enough mathematically to use his opportunities to gain an insight into a mind of remarkable originality. After all, there are mathematicians who believe that Hardy himself struggled hard to have such an insight but with no great success.

He was from all accounts an inspiring teacher, held in great respect and affection by his students, many of whom went on to become fine mathematicians themselves; some among them were leaders on the Indian scene ... S S Pillai become a student of Ananda Rau for whom he cherished life-long affection and respect. ... Luckily for Vijayaraghavan, a real mathematician - Ananda Rau - who could recognise talent that the examination system was incapable of detecting, was at the helm of affairs and he could secure his admission to the B.A. (Honours) course in the University.

R Balasubramanian said, "If Ramanujan's influence in India is the tree … then K Ananda Rau is the root." ... Noted mathematician V Ganapathy Iyer was one of Ananda Rau's students. He commented: "As a student I used to feel that his exposition of any topic was so clear and impressive that I need not study the topic again." C T Rajagopal, a doctoral student of Prof Ananda Rau, said that Ananda Rau's way of working with research scholars was rather novel. He encouraged them and expected them to formulate their own problems. Later, he would discuss the problems with them. In this way, he brought out the best in them. Special emphasis is being given to this quality of Ananda Rau, because many of the stalwarts of the number theory school of South India were mentored by him. Starting from T Vijayaraghavan, S S Pillai, K Chandrasekharan to C T Rajagopal, all were his students. All of them made rich contributions in developing the school of research on number theory in India.

Ananda Rau's early work was done in Cambridge. He was a student of Hardy, and there is no mistaking the influence of Hardy and Littlewood on this work. Ananda Rau had a critical and independent mind and he was conscious of the vast extent of mathematics and the lack of interest in India in many of its branches. He knew Ramanujan in Cambridge and spoke of him with great admiration, but without a trace of the mysticism or romanticism of many others. On his return to India, he took up a position at Presidency College, where he remained until his retirement. He was in the Indian Educational Service, and was thus better off than most other mathematicians. (The IES was abolished shortly after his appointment.)

Ananda Rau's first successes concerned summability by general Dirichlet series (including power series). He established very general "Tauberian theorems" which had resisted the efforts of some of the best analysts. One of the methods of summation to which he contributed some of his most original ideas, Lambert summability as it is called, is intimately connected with the distribution of prime numbers. In later years, Ananda Rau was occupied with modular functions and the representation of integers as sums of squares. It has always seemed to me a great pity that neither Ramanujan nor Ananda Rau ever came into contact with someone like Erich Hecke when they were young. Hecke was a master at combining hard analysis with the arithmetic behaviour of modular (and automorphic) functions, exactly the two topics central in the work of these two Indians.

Ananda Rau's work is of great depth and elegance. He had a strong influence on his students: Vijayaraghavan, S S Pillai, Ganapathy Iyer, Minakshisundaram, K Chandrasekharan and others. He taught them, besides mathematics, the value of intellectual independence.

K Chandrasekharan naturally took up mathematics and wrote his Ph.D. thesis with Ananda Rau as his thesis advisor. K Chandrasekharan's mathematical interests centred around analysis and analytic number theory. The most well-known and influential persons in mathematics in Madras at that time were Ananda Rau, R Vaidyanathaswamy (at the newly created mathematics department in the Madras University) and Fr Racine at Loyola College. ... Whereas Ananda Rau represented analytic number theory, the other two brought to K Chandrasekharan and other students the awareness of different disciplines.

Ananda Rau used to take very good interest in mathematics education. The author cannot forget his unfailing attendance in the monthly meetings arranged by the Madras Mathematics Club where teachers from colleges and the university mingled and listened to informal lectures. He used to say explicitly that our children are made to keep excessively long school hours, whereas all that may be needed to make good students of them (according to their bent) is no more than three hours of school per day. He also believed that every one who sits for a degree examination should be given a degree, since such a system alone would foster genuine scholarship and justify the ends of the university even if it be that it produced only a few men of outstanding ability. Consistent with these opinions was his argument that an employment agency, whether of the Government or not, should devise its own system of selecting candidates instead of leaving the selection to the university and its system of examinations.

Ananda Rau's life, especially in its second half, had an undertone of silent suffering and dedicated studiousness, relieved though by his hobby of gardening pursued with more than amateur skill. Death very nearly claimed him after a kidney operation in November 1936, while it actually claimed two of his dear ones at different times. His wife (whom he had married before he left for Cambridge) passed away in 1928 and one of his daughters in 1940. So it was his mother who, till her death in 1958, became largely responsible for the care of his home and of his surviving three daughters (the first and the two last) as long as they needed it. In the closing years of his life, when he became blind of one eye and physical disabilities began to weigh upon him increasingly, a constant preoccupation with him was numbers expressible as specific sums of squares. During this period, his only personal contacts were with his family and with his pupils in the Ramanujan Institute and their own pupils. Many in the Ramanujan Institute will long remember his tall spare figure, worn away so thin as to seem but a vehicle of shining thought.