Work
Most of his 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 also wrote later in his book A mathematician’s miscellany that his actually only rediscovered result did not shed a bright light on the isolated state 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 of dynamical systems.
He is also remembered for his book of reminiscences, A Mathematician's Miscellany (new edition published in 1986).
Among his own Ph. D. students were Sarvadaman Chowla, Harold Davenport and Donald C. Spencer. Spencer reported that in 1941 when he (Spencer) was about to get on the boat that would take him home to the United States, Littlewood reminded him: "n, n alpha, n beta!" (referring to Littlewood's conjecture).
His collaborative work, carried out by correspondence, covered fields in Diophantine approximation and Waring's problem, in particular. In his other work Littlewood 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 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 inequality on bilinear forms was a forerunner of the later Grothendieck tensor norm theory.
Read more about this topic: John Edensor Littlewood
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