Harold Davenport
Encyclopedia
Harold Davenport FRS (30 October 1907 – 9 June 1969) was an English mathematician
, known for his extensive work in number theory
.
, Accrington
, Lancashire, he was educated at Accrington
Grammar School, the University of Manchester
, where he graduated in 1927, and Trinity College, Cambridge
. He became a research student of J. E. Littlewood, working on the question of the distribution of quadratic residues.
s, for the particular case of some special hyperelliptic curves such as
Bounds for the zeroes of the local zeta-function immediately imply bounds for sums
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 sum
s, and contact with Hans Heilbronn
, with whom Davenport would later collaborate. In fact, as Davenport later admitted, his inherent prejudices against algebraic methods ("what can you do with algebra?") probably limited the amount he learned, in particular in the "new" algebraic geometry
and Artin
/Noether
approach to abstract algebra
.
in 1937, just at the time when Louis Mordell
had recruited émigrés from continental Europe to build an outstanding department. He moved into the areas of diophantine approximation
and geometry of numbers
. 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
.
He was President of the London Mathematical Society
from 1957 to 1959. After professorial positions at the University of Wales
and University College London
, he was appointed to the Rouse Ball Chair of Mathematics in Cambridge in 1958. There he remained until his death, of lung cancer.
. James is Hebron and Medlock Professor of Information Technology at the University of Bath
.
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. This implied problem-solving, and hard-analysis methods. 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
.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, known for his extensive work in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
.
Early life
Born in HuncoatHuncoat
Huncoat is a small village in Lancashire, England; situated in the North West. It is located to the east of Accrington.Huncoat railway station is on the East Lancashire Line.-Origins:...
, Accrington
Accrington
Accrington is a town in Lancashire, within the borough of Hyndburn. It lies about east of Blackburn, west of Burnley, north of Manchester city centre and is situated on the mostly culverted River Hyndburn...
, Lancashire, he was educated at Accrington
Accrington
Accrington is a town in Lancashire, within the borough of Hyndburn. It lies about east of Blackburn, west of Burnley, north of Manchester city centre and is situated on the mostly culverted River Hyndburn...
Grammar School, the University of Manchester
University of Manchester
The University of Manchester is a public research university located in Manchester, United Kingdom. It is a "red brick" university and a member of the Russell Group of research-intensive British universities and the N8 Group...
, where he graduated in 1927, and Trinity College, Cambridge
Trinity College, Cambridge
Trinity College is a constituent college of the University of Cambridge. Trinity has more members than any other college in Cambridge or Oxford, with around 700 undergraduates, 430 graduates, and over 170 Fellows...
. He became a research student of J. E. Littlewood, working on the question of the distribution of quadratic residues.
First steps in research
The attack on the distribution question leads quickly to problems that are now seen to be special cases of those on local zeta-functionLocal zeta-function
In number theory, a local zeta-functionis a function whose logarithmic derivative is a generating functionfor the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F.-Formulation:...
s, for the particular case of some special hyperelliptic curves such as
- Y2 = X(X − 1) (X − 2) ... (X − k).
Bounds for the zeroes of the local zeta-function immediately imply bounds for sums
- Σ χ(x(x − 1) (x − 2) ... (x − k)).
where χ is the Legendre symbol
Legendre symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo a prime number p: its value on a quadratic residue mod p is 1 and on a quadratic non-residue is −1....
modulo
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
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
Marburg
Marburg is a city in the state of Hesse, Germany, on the River Lahn. It is the main town of the Marburg-Biedenkopf district and its population, as of March 2010, was 79,911.- Founding and early history :...
and Göttingen
Göttingen
Göttingen is a university town in Lower Saxony, Germany. It is the capital of the district of Göttingen. The Leine river runs through the town. In 2006 the population was 129,686.-General information:...
working with Helmut Hasse
Helmut Hasse
Helmut Hasse was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local classfield theory and diophantine geometry , and to local zeta functions.-Life:He was born in Kassel, and died in...
, an expert on the algebraic theory. This produced the work on the Hasse-Davenport relations for Gauss sum
Gauss sum
In mathematics, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typicallyG := G= \sum \chi\cdot \psi...
s, and contact with Hans Heilbronn
Hans Heilbronn
Hans Arnold Heilbronn was a mathematician.He was born into a German-Jewish family. He was a student at the universities of Berlin, Freiburg and Göttingen, where he met Edmund Landau, who supervised his doctorate...
, with whom Davenport would later collaborate. In fact, as Davenport later admitted, his inherent prejudices against algebraic methods ("what can you do with algebra?") probably limited the amount he learned, in particular in the "new" algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
and Artin
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
/Noether
Emmy Noether
Amalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...
approach to abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
.
Later career
He took an appointment at the University of ManchesterSchool of Mathematics, University of Manchester
The School of Mathematics at the University of Manchester is one of the largest mathematics departments in the United Kingdom, with around 80 academic staff and an undergraduate intake of roughly 400 a year and another 200 postgraduate students...
in 1937, just at the time when Louis Mordell
Louis Mordell
Louis Joel Mordell was a British mathematician, known for pioneering research in number theory. He was born in Philadelphia, USA, in a Jewish family of Lithuanian extraction...
had recruited émigrés from continental Europe to build an outstanding department. He moved into the areas of diophantine approximation
Diophantine approximation
In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....
and geometry of numbers
Geometry of numbers
In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by ....
. These were fashionable, and complemented the technical expertise he had in the Hardy-Littlewood circle method
Hardy-Littlewood circle method
In mathematics, the Hardy–Littlewood circle method is one of the most frequently used techniques of analytic number theory. It is named for G. H. Hardy and J. E...
; he was later, though, to let drop the comment that he wished he'd spent more time on the Riemann hypothesis
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
.
He was President of the London Mathematical Society
London Mathematical Society
-See also:* American Mathematical Society* Edinburgh Mathematical Society* European Mathematical Society* List of Mathematical Societies* Council for the Mathematical Sciences* BCS-FACS Specialist Group-External links:* * *...
from 1957 to 1959. After professorial positions at the University of Wales
University of Wales
The University of Wales was a confederal university founded in 1893. It had accredited institutions throughout Wales, and formerly accredited courses in Britain and abroad, with over 100,000 students, but in October 2011, after a number of scandals, it withdrew all accreditation, and it was...
and University College London
University College London
University College London is a public research university located in London, United Kingdom and the oldest and largest constituent college of the federal University of London...
, he was appointed to the Rouse Ball Chair of Mathematics in Cambridge in 1958. There he remained until his death, of lung cancer.
Personal life
Davenport married Anne Lofthouse, whom he met at the University College of North Wales at Bangor, in 1944; they had two children, Richard and JamesJames Davenport (professor)
James Harold Davenport is a British computer scientist who works in computer algebra. He is the Hebron and Medlock Professor of Information Technology at the University of Bath in Bath, England....
. James is Hebron and Medlock Professor of Information Technology at the University of Bath
University of Bath
The University of Bath is a campus university located in Bath, United Kingdom. It received its Royal Charter in 1966....
.
Influence
From about 1950 he was the obvious leader of a "school", somewhat unusually in the context of British mathematics. The successor to the school of mathematical analysisMathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
of G. H. Hardy
G. H. Hardy
Godfrey Harold “G. H.” Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis....
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. This implied problem-solving, and hard-analysis methods. The outstanding works of Klaus Roth
Klaus Roth
Klaus Friedrich Roth is a British mathematician known for work on diophantine approximation, the large sieve, and irregularities of distribution. He was born in Breslau, Prussia, but raised and educated in the UK. He graduated from Peterhouse, Cambridge in 1945...
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
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
. This concrete emphasis on problems stood in sharp contrast with the abstraction of Bourbaki, who were then active just across the English Channel
English Channel
The English Channel , often referred to simply as the Channel, is an arm of the Atlantic Ocean that separates southern England from northern France, and joins the North Sea to the Atlantic. It is about long and varies in width from at its widest to in the Strait of Dover...
.
Books
- The Higher Arithmetic: An Introduction to the Theory of Numbers (1952)
- Analytic methods for Diophantine equations and Diophantine inequalities (1962)
- Multiplicative number theory (1967)
- The collected works of Harold Davenport (1977) in four volumes, edited by B. J. Birch, H. Halberstam, C. A. Rogers