Hardy came up to Cambridge at a time when profound changes in the mathematical teaching were in progress. The theory of functions of a complex variable, created by Cauchy in 1825, had been the backbone of continental mathematics for seventy years. In England it was virtually unknown till, in the nineties, A R Forsyth gave some post-graduate lectures on it at Cambridge. They were attended by E T (now Sir Edmund) Whittaker, who perceived its cardinal importance and gained for it the status, which it now occupies in all Universities, of being the principal branch of Pure Mathematics studied by undergraduates in their final year. The theory of functions of a real variable, also of continental origin, was introduced soon afterwards by E W Hobson. Both the complex and the real variable were subjects after Hardy's own heart. He had a strong feeling for the elegance of the formulae of the former, and an equally keen realisation of the need to establish them with the rigour demanded in real variable proofs. Rigorous investigation of the conditions in which a formula is true was to form the subject of an enormous number of his papers. They include the best of his early papers, and continued all through his life.
Hardy's most vital work was, however, in a different field. In approaching it the biographer is met by a great difficulty. Almost all of this work was done in collaboration. No other mathematician has collaborated so much or so fruitfully. In 1914 he wrote two brilliant papers, one proving that Riemann's zeta function has an infinity of zeros on the critical line, the other on the convexity of the mean value of the modulus of an analytic function. Apart from these none of his very numerous papers can bear comparison with the best work of Hardy and J E Littlewood, or of Hardy and S Ramanujan. One is impressed by their elegance, their learning, their usefulness, their technique, but they lack the boldness of conception which is so evident in the best of his joint work. The papers of Hardy and Littlewood cover many fields, especially Tauberian theorems, Fourier series and problems of partitio numerorum. In the latter field, in which he also collaborated with Ramanujan, his work is universally regarded as one of the highest achievements of the century in mathematics. The great work of Hadamard in the last decade of the nineteenth century had added a new chapter, a kind of physical geography, to complex variable theory, by showing that the main features of functions, their zeros, rates of growth, and so forth, were connected in a manner much closer than had been suspected; and he had made a brilliant application of this discovery to prove the prime number theorem for the first time. Hardy and Littlewood applied this new conception of complex variable theory to problems even more ancient and intractable. Such is the conjecture that every even number is the sum of two primes. This was asserted by Goldbach in 1742, but no progress whatever had been made towards establishing it till Hardy and Littlewood brought their "circle method" to bear on it. They did not, indeed, solve the problem but they made a great step in advance, and Russian developments of this work leave little to be done. Still more complete was the work of Hardy and Ramanujan on partitions, culminating in an astonishingly accurate approximate formula for p(n).
Hardy was elected an Honorary Fellow of the Society in 1946.
Godfrey Harold Hardy's RSE obituary by J M Whittaker appeared in Royal Society of Edinburgh Year Book 1948 and 1949, 24-25.
See also Obituary Notices of Fellows of the Royal Society, vi, 1948-49, pp. 447-462.