Harold Davenport

Davenport in 1968 Born (1907-10-30)30 October 1907

Huncoat, Lancashire, England Died 9 June 1969(1969-06-09) (aged 61)

Cambridge, England Alma mater University of Manchester Trinity College, Cambridge Known for

Children James H. Davenport Awards Rayleigh Prize (1930) Adams Prize (1940) Senior Berwick Prize (1954) Sylvester Medal (1967) Fellow of the Royal Society [1] • Scientific career Fields Number theory Institutions University of Manchester University of Wales University College London University of Cambridge Doctoral advisor John Edensor Littlewood [2] Doctoral students John Horton Conway Alan Baker Peter Elliott H. L. Montgomery Martin Huxley G. L. Watson [2]

Harold Davenport FRS [1] (30 October 1907 – 9 June 1969) was an English mathematician whose contributions to the field of number theory were both profound and prolific. His work spanned several decades, leaving an indelible mark on the analytic and diophantine branches of the discipline. His intellectual journey, marked by numerous accolades including the Rayleigh Prize in 1930, the Adams Prize in 1940, the Senior Berwick Prize in 1954, and the prestigious Sylvester Medal in 1967, solidified his reputation as a leading figure in 20th-century mathematics.

Davenport’s legacy is further cemented by the various mathematical concepts that bear his name, illustrating the breadth and depth of his influence. These include the Davenport–Erdős theorem, a fundamental result in probabilistic number theory concerning the normal order of arithmetic functions; Davenport–Schinzel sequences, which found unexpected applications in computational geometry and combinatorics; the Davenport–Schmidt theorem, a crucial development in diophantine approximation; and the Hasse–Davenport relations, which connect different types of Gauss sums, bridging algebraic and analytic number theory. His academic lineage is equally distinguished, having been a doctoral student of John Edensor Littlewood and subsequently advising a formidable cohort of mathematicians, including luminaries like John Horton Conway and Alan Baker, among others.

Early life and education

Born on 30 October 1907, in the unassuming locale of Huncoat, Lancashire, England, Harold Davenport embarked on a path that would lead him to the pinnacles of mathematical thought. His formative education took place at Accrington Grammar School, a stepping stone that prepared him for the rigors of higher academia. He then matriculated at the University of Manchester, where he distinguished himself, graduating in 1927. This academic foundation was further strengthened at the venerable Trinity College, Cambridge, a bastion of mathematical excellence.

It was at Cambridge that Davenport commenced his doctoral studies under the tutelage of the eminent mathematician John Edensor Littlewood [2]. His early research focused on a particularly intricate problem within number theory: the distribution of quadratic residues. This initial foray into the complexities of number theory would set the trajectory for his illustrious career, laying the groundwork for his later, more expansive contributions. One might almost say he chose a rather specific corner of the universe to scrutinize, and then proceeded to dissect it with surgical precision.

First steps in research

The initial investigation into the distribution of quadratic residues quickly propelled Davenport into a domain of problems that are now recognized as specialized instances of those concerning local zeta-functions. Specifically, his work touched upon the particular manifestations arising from certain special hyperelliptic curves. Consider, for instance, a curve defined by the equation:

$Y^{2}=X(X-1)(X-2)\ldots (X-k)$

Such equations, seemingly benign, conceal profound mathematical challenges. The establishment of effective bounds for the zeroes of these local zeta-functions directly yields crucial implications for bounding sums of the form:

$\sum \chi (X(X-1)(X-2)\ldots (X-k))$

In this expression, $\chi$ represents the Legendre symbol modulo a prime number $p$ , and the summation extends over a complete set of residues modulo $p$ . Understanding these sums is pivotal for grasping the subtle patterns within number sequences.

Recognizing the crucial interconnectedness of these concepts, Davenport, having secured a prestigious Trinity research fellowship, spent a significant period in 1932–1933 in Germany, specifically in Marburg and Göttingen. There, he collaborated with Helmut Hasse, a preeminent authority in the realm of algebraic theory. This collaboration proved to be exceptionally fruitful, culminating in the foundational work on the Hasse–Davenport relations for Gauss sums. This period also fostered his connection with Hans Heilbronn, with whom he would later forge further productive collaborations.

However, even in the pursuit of groundbreaking insights, human predispositions persist. Davenport himself later conceded that his inherent skepticism towards purely algebraic methodologies—a sentiment he pithily encapsulated as "what can you do with algebra?"—might have inadvertently constrained the depth of his learning during this critical period. This skepticism, perhaps a testament to the prevailing analytical traditions he was accustomed to, likely limited his engagement with the burgeoning "new" algebraic geometry and the revolutionary Artin/Noether approach to abstract algebra. A rather predictable human flaw, if you ask me, to cling to the familiar even when the unfamiliar offers new vistas.

Beyond these theoretical advancements, Davenport also delved into more concrete, albeit equally challenging, problems. In 1946, he definitively proved that 8436 stands as the largest tetrahedral number that can be expressed in the form $2^{a}+3^{b}+1$ for some non-negative integers $a$ and $b$ . Not content with that, in 1947, he demonstrated that 5040 represents the largest factorial of the form $n(n+4)(n+6)$ for some integer $n$ . These solutions were not achieved through mere guesswork but necessitated the deployment of advanced techniques, including the sophisticated Brun sieve and other specialized methods within number theory. A testament to the fact that even seemingly simple questions can hide immense complexity, requiring an almost perverse dedication to unravel.

Later career

Davenport's professional journey continued its upward trajectory when he accepted an appointment at the University of Manchester in 1937. This was a particularly dynamic period for the department, as Louis Mordell had been instrumental in recruiting a cohort of brilliant émigrés from continental Europe, effectively transforming Manchester into an outstanding center for mathematical research. This environment provided fertile ground for Davenport's talents, and he soon expanded his research focus into the interconnected realms of diophantine approximation and the geometry of numbers.

These fields were not only intellectually stimulating but also highly fashionable within the mathematical community at the time. They proved to be a natural complement to his already formidable technical expertise in the Hardy–Littlewood circle method, a powerful analytic technique for tackling problems in additive number theory. Despite his success in these areas, a fleeting comment he later made revealed a subtle undercurrent of regret: he wished he had dedicated more of his time to the elusive Riemann hypothesis. A sentiment many mathematicians share, I imagine, peering into the abyss of unsolved problems and wondering what could have been.

His leadership qualities were formally recognized when he served as President of the London Mathematical Society from 1957 to 1959 [3], a period during which he undoubtedly influenced the direction of mathematical discourse in Britain. After holding distinguished professorial positions at the University of Wales and University College London, Davenport returned to Cambridge in 1958, accepting the prestigious Rouse Ball Chair of Mathematics. He remained in this esteemed position, contributing tirelessly to the academic life of the university, until his passing on 9 June 1969, at the age of 61, due to lung cancer.

Personal life

Beyond the intricate world of numbers and theorems, Harold Davenport maintained a personal life that intertwined with his academic pursuits. In 1944, he married Anne Lofthouse, whom he had met at the University College of North Wales at Bangor. Their union produced two children: Richard and James. The latter, James H. Davenport, followed in his father's academic footsteps, carving out his own distinguished career as the Hebron and Medlock Professor of Information Technology at the University of Bath, demonstrating that intellectual rigor, like certain genetic predispositions, sometimes runs in families.

Influence

From approximately 1950 onwards, Harold Davenport emerged as the undeniable leader of a distinct mathematical "school" within Britain. This was a notable occurrence, given the often diffuse nature of academic influence. This school, in many respects, represented a direct successor to the powerful tradition of mathematical analysis established by figures such as G. H. Hardy and J. E. Littlewood. However, Davenport's school exhibited a more focused specialization, primarily devoted to number theory, and more specifically, to its analytic side, which had experienced a vibrant flourishing throughout the 1930s.

The defining characteristic of this approach was an unwavering emphasis on concrete problem-solving and the rigorous application of hard-analysis methods. This methodology, demanding both ingenuity and technical mastery, proved exceptionally effective, as exemplified by the groundbreaking works of Klaus Roth and Alan Baker in the demanding field of diophantine approximation. The philosophical underpinnings of this school were succinctly captured by two oft-quoted maxims attributed to Davenport: "the problems are there," a stark reminder of the inherent challenges awaiting discovery, and "I don't care how you get hold of the gadget, I just want to know how big or small it is," which underscored a pragmatic, results-oriented approach to mathematical tools and estimations. These sentiments, focused on the tangible and measurable, resonate even today within disciplines like combinatorics, where practical outcomes often supersede abstract elegance.

This concrete, problem-driven emphasis stood in sharp contrast to the highly abstract and formalist movement championed by Nicolas Bourbaki, whose collective work was actively shaping mathematical thought just across the English Channel during the same period. While Bourbaki sought to rebuild mathematics from its foundational axioms, Davenport's school remained firmly rooted in the pursuit of solutions to specific, challenging problems, demonstrating that diverse philosophical approaches can coexist and contribute to the advancement of mathematics. One could argue it was less about how the universe was constructed, and more about how to navigate its more inconvenient corners.

Books

Harold Davenport authored several influential books that served as essential texts for students and researchers in number theory and related fields. These works are still highly regarded for their clarity, rigor, and comprehensive treatment of complex topics.

The Higher Arithmetic: An Introduction to the Theory of Numbers (1952) [5]. This book introduced countless students to the fundamental concepts and methods of number theory, providing a clear and accessible entry point into a notoriously challenging field. It remains a classic for its pedagogical excellence.
Analytic methods for Diophantine equations and Diophantine inequalities (1962). A second edition, revised by T. D. Browning, was published in 2005 by Cambridge University Press, bearing the ISBN 0-521-60583-0 [6]. This work delved into the powerful analytic techniques used to tackle problems involving integer solutions to polynomial equations and inequalities, a core area of Davenport's expertise.
Multiplicative number theory (1967) [7]. This book became a cornerstone for advanced studies in the field, exploring the intricate properties of multiplicative functions and their distribution. A second edition, thoughtfully revised by Hugh L. Montgomery, further extended its reach and relevance.
The collected works of Harold Davenport (1977). This monumental four-volume collection, meticulously edited by B. J. Birch, H. Halberstam, and C. A. Rogers [8], stands as a comprehensive testament to his enduring contributions, compiling his vast array of research papers and offering a complete overview of his mathematical legacy. It's a rather hefty reminder of how much one person can accomplish if they simply refuse to be distracted by the trivialities of existence.