Most of Littlewood's work was in the field of mathematical analysis. He began research under the supervision of Ernest William Barnes, who suggested that he attempt to prove the Riemann hypothesis: Littlewood showed that if the Riemann hypothesis is true then the prime number theorem follows and obtained the error term. This work won him his Trinity fellowship. However, the link between the Riemann hypothesis and the prime number theorem had been known before in Continental Europe, and Littlewood wrote later in his book, A Mathematician's Miscellany that his rediscovery of the result did not shed a positive light on the isolated nature of British mathematics at the time. He coined Littlewood's law, which states that individuals can expect "miracles" to happen to them, at the rate of about one per month. He continued to write papers into his eighties, particularly in analytical areas of what would become the theory ofdynamical systems. Littlewood is also remembered for his book of reminiscences, A Mathematician's Miscellany. Among his own PhD students were Sarvadaman Chowla, Harold Davenport, and Donald C. Spencer. Spencer reported that in 1941 when he was about to get on the boat that would take him home to the United States, Littlewood reminded him: "n, n alpha, n beta!". Littlewood's collaborative work, carried out by correspondence, covered fields in Diophantine approximation and Waring's problem, in particular. In his other work, he collaborated with Raymond Paley on Littlewood–Paley theory in Fourier theory, and with Cyril Offord in combinatorial work on random sums, in developments that opened up fields that are still intensively studied. He worked with Mary Cartwright on problems in differential equations arising out of early research on radar: their work foreshadowed the modern theory of dynamical systems. Littlewood's 4/3 inequality on bilinear forms was a forerunner of the later Grothendiecktensor norm theory.
With Hardy
Littlewood collaborated for many years with G. H. Hardy. Together they devised the first Hardy–Littlewood conjecture, a strong form of the twin prime conjecture, and the second Hardy–Littlewood conjecture. He also, with Hardy, identified the work of the Indian mathematicianSrinivasa Ramanujan as that of a genius, and supported him in travelling to the UK and working at Cambridge. A self-taught mathematician, Ramanujan later became a Fellow of the Royal Society, Fellow of Trinity College, Cambridge, and widely recognised as on a par with other geniuses such as Euler and Jacobi. In a 1947 lecture, the Danish mathematician Harald Bohr said, "To illustrate to what extent Hardy and Littlewood in the course of the years came to be considered as the leaders of recent English mathematical research, I may report what an excellent colleague once jokingly said: 'Nowadays, there are only three really great English mathematicians: Hardy, Littlewood, and Hardy–Littlewood.'" There is a story that at a conference Littlewood met a German mathematician who said he was most interested to discover that Littlewood really existed, as he had always assumed that Littlewood was a name used by Hardy for lesser work which he did not want to put out under his own name; Littlewood apparently roared with laughter. There are versions of this story involving both Norbert Wiener and Edmund Landau, who, it is claimed, "so doubted the existence of Littlewood that he made a special trip to Great Britain to see the man with his own eyes".
Cultural references
John Littlewood is depicted in two films covering the life of Ramanujan – Ramanujan in 2014 portrayed by Michael Lieber and The Man Who Knew Infinity in 2015 portrayed by Toby Jones.
