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.
Theory of the distribution of prime numbers In 1914, Littlewood published his first result in the field of
analytic number theory concerning the error term of the
prime-counting function. If \pi(x) denotes the number of primes up x, then the
prime number theorem implies that \pi(x)\sim\operatorname{Li}(x), where \operatorname{Li}(x) is the
offset logarithmic integral. Numerical evidence seemed to suggest that \pi(x) for all x. Littlewood, however proved that the difference \pi(x)-\operatorname{Li}(x) changes sign infinitely often.
Collaboration with G. H. 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.
Ramanujan He also, with Hardy, identified the work of the Indian mathematician
Srinivasa Ramanujan as that of a genius and supported him in travelling from India to work 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.
Collaboration with Mary Cartwright In the late 1930s, as the prospect of war loomed, the
Department of Scientific and Industrial Research sought the interest of pure mathematicians in the properties of
non linear differential equations that were needed by radio engineers and scientists. The problems appealed to Littlewood and
Mary Cartwright, and they worked on them independently during the next 20 years. The problems that Littlewood and Cartwright worked on concerned
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
Grothendieck tensor norm theory. ==Military service WWI – ballistics work==