VI.73 Godfrey Harold Hardy
b. Cranleigh, England, 1877; d. Cambridge, 1947
Number theory; analysis
G. H. Hardy was the most influential mathematician in Britain in the twentieth century. With the exception of the years from 1919 to 1931, when he was the Savilian professor of geometry in Oxford, he spent his adult life in Cambridge, where from 1931 until his retirement in 1942 he was the Sadleirian professor of pure mathematics. He became a Fellow of the Royal Society in 1910 and was awarded a Royal Medal in 1920 and the Sylvester Medal in 1940. He died on the day the Royal Society’s highest honor, the Copley Medal, was to be presented to him.
At the beginning of the twentieth century, the standard of mathematical analysis was rather low in Britain; Hardy did much to remedy this situation, not only through his research, but also by publishing A Course of Pure Mathematics in 1908. This book, which he wrote as “a missionary talking to cannibals,” had a tremendous influence on several generations of mathematicians in the United Kingdom. Unfortunately, Hardy’s love of pure mathematics, and analysis in particular, somewhat stifled the growth of applied mathematics and algebraic subjects for several decades.
In 1911 he began a long collaboration with LITTLE-WOOD [VI.79], with whom he wrote almost one hundred papers: this partnership is generally considered to have been the most fruitful in the history of mathematics. They worked on convergence and summability of series, inequalities, ADDITIVE NUMBER THEORY [V.27] (including Waring’s problem and Goldbach’s conjecture), and Diophantine approximation.
Hardy was one of the first to do important work on the RIEMANN HYPOTHESIS [IV.2 §3] when, in 1914, he proved that the zeta function ζ(s) ζ(σ + it) has infinitely many zeros on the critical line σ = 
(see LITTLEWOOD [VI.79]). Later, with Littlewood, he proved deep extensions of this result.
From 1914 to 1919 he collaborated with the largely self-taught Indian genius, SRINIVASA RAMANUJAN [VI.82]. They wrote five papers, the most famous of which is about p(n), the number of partitions of n.
This is a rapidly growing function: p(5) = 7 but
p (200) = 3 972 999 029 388.
The GENERATING FUNCTION [IV.18 §§2.4, 3] of p (n), that is,
where Γ is a circle about the origin of radius just less than 1. In 1918, Hardy and Ramanujan not only gave a rapidly convergent asymptotic formula for p(n) but also showed that, for n large enough, p (n) could be calculated exactly by taking the integer nearest to the sum of the first few terms. In particular, p (200) can be computed from the first five terms.
Hardy and Ramanujan proved their asymptotic formula for p (n) with the aid of the “circle method”; later, Hardy and Littlewood developed this method into one of the most powerful tools in analytic number theory. In order to estimate contour integrals like the one above, Hardy and Littlewood found it advisable to break up the circle of integration in a subtle way.
Another Hardy-Ramanujan result concerns the number ω(n) of distinct prime divisors of a “typical” number n. They proved that a “typical” number n has about log log n distinct prime factors in a certain precise sense. In 1940 Erd
s and Kac sharpened and extended this result by showing that additive number-theoretic functions like ω(n) obey the GAUSSIAN LAW [III.71 §5] of errors: this gave birth to the important field of probabilistic number theory.
Hardy’s name has been attached to several concepts and results, including Hardy spaces, Hardy’s inequality, and the HARDY-LITTLEWOOD MAXIMAL THEOREM [IV.11 §3]. For 0 < P ≤ ∞ the Hardy space HP consists of functions analytic in the unit disk that are bounded in various ways; in particular, H” consists of bounded analytic functions. Hardy and Littlewood deduced fundamental properties of HP from their maximal theorem, which relates a function to its “radial limits” at the boundary of the disk. The theory of HP spaces has found numerous applications not only in analysis, but also in probability theory and control theory.
Hardy and Littlewood loved inequalities of all kinds; their book on the subject with George Pólya, an instant classic the moment it was published in 1934, greatly influenced the development of hard analysis.
Although Hardy was fiercely proud of the purity of his mathematics, in a paper published in 1908 he formulated the extension of the Mendelian law about the proportions of dominant and recessive characters. This law, which later became known as the Hardy-Weinberg law, refuted the idea “that a dominant character should show a tendency to spread over a whole population, or that a recessive should tend to die out.” In a later article he dealt a severe blow to eugenics by giving a simple mathematical argument that showed the futility of forbidding people with “undesirable” characteristics to breed.
In his interest in mathematical philosophy, Hardy was a disciple of RUSSELL [VI.71], whose political views he also shared. He was a secretary of the committee which forced the abolition of the order of merit in the Mathematical Tripos through a reluctant Senate in 1910, and many years later he fought hard for the abolition (not reform!) of the Mathematical Tripos itself, which he considered to be harmful to mathematics in the United Kingdom. After World War I, Hardy led the British efforts to heal the wounds of the international mathematical community, and with the advent of the Nazi persecutions on the Continent in the early 1930s, he was an important figure in an extensive network finding jobs for refugee mathematicians in the United States, Britain, and the Commonwealth. He was a great supporter of the London Mathematical Society: he was not only one of the secretaries for close to twenty years, but also its president for two terms.
Hardy was a militant atheist; as an affectation, he liked to talk of God as his personal enemy. He was a great conversationalist, and was fond of various intellectual games, like putting together cricket teams of bores, bogus poets, Fellows of a Cambridge college, and so on. He loved ball games, especially cricket, baseball, bowls (with the curved woods of his college), and real tennis (as opposed to lawn tennis); to praise people, he frequently likened them to outstanding cricketers.
He had an exceptional gift for collaboration and launching young mathematicians on their research careers. He was a master not only of mathematics, but also of English prose; he was lively and charming, and left a lasting impression even on his casual acquaintances. His poetic book A Mathematician’s Apology, written toward the end of his life, gives a rare insight into the world of a mathematician.
Further Reading
Hardy, G. H. 1992. A Mathematician’s Apology, with a foreword by C. P. Snow. Cambridge: Cambridge University Press. (Reprint of the 1967 edition.)
Hardy, G. H., J. E. Littlewood, and G. Pólya. 1988. Inequalities. Cambridge: Cambridge University Press. (Reprint of the 1952 edition.)