Harold Davenport FRS[1] (30 October 1907 – 9 June 1969) was an English mathematician, known for his extensive work in number theory.

Born on 30 October 1907 in Huncoat, Lancashire, Davenport was educated at Accrington Grammar School, the University of Manchester (graduating in 1927), and Trinity College, Cambridge.

He became a research student of John Edensor Littlewood,[2] working on the question of the distribution of quadratic residues.

, where χ is the Legendre symbol modulo a prime number p, and the sum is taken over a complete set of residues mod p. In the light of this connection it was appropriate that, with a Trinity research fellowship, Davenport in 1932–1933 spent time in Marburg and Göttingen working with Helmut Hasse, an expert on the algebraic theory.

This produced the work on the Hasse–Davenport relations for Gauss sums, and contact with Hans Heilbronn, with whom Davenport would later collaborate.

He took an appointment at the University of Manchester in 1937, just at the time when Louis Mordell had recruited émigrés from continental Europe to build an outstanding department.

These were fashionable, and complemented the technical expertise he had in the Hardy–Littlewood circle method; he was later, though, to let drop the comment that he wished he'd spent more time on the Riemann hypothesis.

The successor to the school of mathematical analysis of G. H. Hardy and J. E. Littlewood, it was also more narrowly devoted to number theory, and indeed to its analytic side, as had flourished in the 1930s.

The outstanding works of Klaus Roth and Alan Baker exemplify what this can do, in diophantine approximation.

Two reported sayings, "the problems are there", and "I don't care how you get hold of the gadget, I just want to know how big or small it is", sum up the attitude, and could be transplanted today into any discussion of combinatorics.

This concrete emphasis on problems stood in sharp contrast with the abstraction of Bourbaki, who were then active just across the English Channel.