# Harold Davenport

### Quick Info

Huncoat, Lancashire, England

**Harold Davenport**worked on number theory, in particular the geometry of numbers, Diophantine approximation and the analytic theory of numbers. He wrote a number of important textbooks and monographs including

*The higher arithmetic*(1952).

### Biography

**Harold Davenport**'s parents were Nancy Barnes and Percy Davenport. Nancy was the daughter of John Barnes who owned Perseverance Mill, and Percy worked for John Barnes, first as a clerk and later as the company secretary. Nancy and Percy had two children, Harold having a younger sister Grace.

Harold attended Accrington Grammar School from the age of about ten years. His main interests were mathematics and chemistry [7]:-

He was much inspired by his chemistry master, Mr Ackroyd, and by his mathematics mistress, Miss Heap, 'a lady with enthusiasm for mathematics, who paid no attention - thank God - to any regular syllabus or curriculum there may have been'.There were other subjects which interested Harold too, for he loved reading and read every Dickens classic that he could borrow from the local Public Library.

In 1924 he obtained two scholarships to attend Manchester University, one from Lancashire County and a university scholarship. He studied mathematics and chemistry at Manchester being taught complex analysis by Mordell and applied mathematics by Milne. At first he seriously considered a career in chemistry, and he wanted to take both subjects to honours level. But slowly he became more certain that his future was with mathematics so, when he found he could not continue to study both subjects, the decision was not too difficult. He graduated from Manchester with First Class honours in mathematics in 1927.

After Manchester he went to Trinity College, Cambridge, to take another 'first degree' which was a common thing to do at that time, and he was advised to do so by Milne. Among the friends he made at Cambridge were Coxeter, Paley, Sadler, and Ursell. He had Fowler as an applied mathematics director, and at first his pure mathematics director was S Pollard, then later was Besicovitch. Coxeter wrote (see for example [7]):-

When Davenport was working for the Tripos he seemed wonderfully relaxed. He would give me a cheerful welcome whenever I dropped in to see him under the Clock in Trinity Great Court. I would find him listening to Scheherazade on the phonograph or reading Gibbon's 'Decline and Fall' for the third time. He and Sadler and I often went together to a cinema or to the Festival Theatre. I once asked him how he managed to do such a prodigious amount of mathematics. He replied: "Between midnight and 3 a.m." His mind must have worked so rapidly that he could do in those three hours more than anyone else could do in six.Davenport was most attracted by Littlewood's lectures on the theory of primes and those of Besicovitch on almost periodic functions. Davenport wrote a Ph.D. thesis at Cambridge under Littlewood's supervision. His research involved studying the distribution of quadratic residues, and he invented new methods to attack his problems involving character sums and exponential sums.

He was awarded a Rayleigh Prize in 1930 but expressed disappointment at not being a Smith's Prizeman. He was elected to a Trinity fellowship in 1932 and soon after taking up the fellowship he visited Hasse in Marburg and wrote an important joint paper with him. This visit came about because Hasse wished to learn English and had written to Mordell asking him to recommend a young English mathematician who might come to Göttingen to teach him. In fact not only did Davenport teach Hasse English, but he himself became fluent in German during his year at Göttingen [2]:-

... he used to claim that Germans would accept him as a compatriot albeit with the accent of a distant province. (To the Anglo-Saxon ear that province indubitably lay near the Mersey.)Davenport met Heilbronn while in Germany and they began a research collaboration which lasted for many years. After returning to Cambridge his research struck an incredibly rich vein and he published a great number of papers. At this time life in Cambridge was enriched by a large number of visiting mathematicians who were escaping from the Nazi threat on the Continent. Those who interacted with Davenport included Richard Rado, Hirsch, Courant, Taussky (later Taussky-Todd), Kober and Mahler.

He left Cambridge in 1937, accepting an offer from Mordell of an assistant lectureship at the University of Manchester. There he was influenced by Mordell to become interested in both the geometry of numbers and Diophantine approximation. While he taught at Manchester he had a number of outstanding colleagues including Mahler, Erdős, and Beniamino Segre. During his four years on the staff he received a number of honours including a fellowship of the Royal Society and the Adams Prize from the University of Cambridge, both in 1940.

In 1941 Davenport was appointed to the chair of mathematics at the University College of North Wales at Bangor. Three year later he married Anne Lofthouse who was on the staff of the modern languages department. They had two children, James and Richard. Harold and Anne left Wales and moved to London in 1945 when Davenport succeeded Jeffery as Astor professor of mathematics in University College, London. Rogers writes about the time shortly after Davenport came to University College (see [7]):-

... I attended Davenport's lectures and seminars, and I am proud to claim to be one of Davenport's students. He gave me inspiration and unlimited help, and friendship. At this time Davenport worked mainly on the geometry of numbers and on Diophantine approximation; he also acquired a lasting interest in problems of packing and covering. It was this last interest that spurred me to some of my most satisfying work.Stanford University in California provided an excellent place for study leave during 1947-48 and the friendships he made with Pólya and Szego lasted throughout their lives. In 1958 Davenport returned to Cambridge as Rouse Ball Professor of Mathematics on the retirement of Besicovitch. This gave him less administrative duties, and more opportunities to make visits to other universities such as Göttingen, Ann Arbor, Boulder, and Milan. His style of doing mathematics at Cambridge is described by Lewis in [6]:-

Davenport used to sit from 10 to 12 most mornings drinking coffee and talking to his students and colleagues, including the many post-doctoral visitors who appeared at Cambridge each year. A pad of paper was readily at hand. The students always knew where he could be found and that he was always ready to discuss their latest successes and failures. Usually it was a conversation between him and one other; but the students all sat around waiting their turn to put a question. The conversation was almost entirely mathematical ...Davenport worked on number theory, in particular the geometry of numbers, Diophantine approximation and the analytic theory of numbers. He contributed to Waring's problem early in his career by proving that every sufficiently large number was the sum of sixteen fourth powers. He wrote a number of important textbooks and monographs.

*The higher arithmetic*(1952) was a book written at a low level in an attempt to bring results in number theory before as wide an audience as possible.

At the most advanced level he wrote a monograph

*Analytic methods for Diophantine equations and Diophantine inequalities*(1962) which includes many of his contributions extending the Hardy-Littlewood method. He also wrote an important monograph on the analytic approach to the theory of the distribution of primes

*Multiplicative number theory*(1967).

We mentioned above that Davenport had been elected a Fellow of the Royal Society while still an assistant lecturer. In 1967 the Society awarded him its Sylvester Medal:-

... in recognition of his many distinguished contributions to the theory of numbers.He was President of the London Mathematical Society during 1957-59, and was awarded the Berwick Prize by that Society in 1954. He was elected to the Royal Society of Science of Uppsala in 1964.

Davenport described his philosophy of mathematics in the following way:-

Mathematicians are extremely lucky, they are paid for doing what they would by nature have to do anyway. One should not have a non-teaching fellowship too long, there comes a time when one must make a contribution to society. Great mathematics is achieved by solving difficult problems not by fabricating elaborate theories in search of a problem.Always a heavy smoker (he tried to give up the habit several times but always failed), Davenport succumbed to lung cancer at a young age. His influence on those around him is summed up in [7] as follows:-

... the extent which he helped others can only be guessed, he was probably responsible for encouraging work at least as extensive as his own. ... He made his collaborators and colleagues his friends, and gave them generously of his humour and wisdom. He made a practice of writing helpful letters to all who approached him on mathematical matters whether they were professionals, students, amateurs or even cranks. By correspondence and by direct contact he stimulated and encouraged many mathematicians to do much of their best mathematics.Davenport's character is described in [2]:-

Davenport was a natural conservative. "All change is for the worse" he used to say with complete conviction. He was entirely out of sympathy with the waves of change in the teaching of mathematics but accepted them as an inevitable evil. Selective in his enjoyment of modern technology, he never entered an aeroplane, would use a lift if no alternative existed (at the International Congress in Moscow he trudged up and down the interminable stairs of Stalin's skyscraper), and preferred to send his papers for publication written in his characteristically neat hand.

### References (show)

- H Halberstam, Biography in
*Dictionary of Scientific Biography*(New York 1970-1990). See THIS LINK. - Obituary in
*The Times*See THIS LINK. - L J Mordell, Harold Davenport,
*Acta Mathematica***18**(1971), 1-4. - L J Mordell, Some aspects of Davenport's work,
*Acta Mathematica***18**(1971), 5-11. - Papers by Harold Davenport,
*Acta Arith.***18**(1971), 19-28. - C A Rogers, B J Birch, H Halberstam and D A Burgess, Harold Davenport,
*Bull. London Math. Soc.***4**(1972), 66-99. - C A Rogers, D A Burgess, H Halberstam and B J Birch, Harold Davenport,
*Biographical Memoirs of Fellows of the Royal Society of London***17**(1971), 159-192. - C A Rogers, A brief survey of the work of Harold Davenport,
*Acta Arith.***18**(1971), 13-17.

### Additional Resources (show)

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Written by J J O'Connor and E F Robertson

Last Update October 2003

Last Update October 2003