**Hendrik Kloosterman**was the son of Bindert Hendriks Kloosterman (1857-1935), a structural engineer, and Marijke Durks van der Molen (1861-1936). Bindert and Marijke were married at Achtkarspelen on 1 July 1899. For Marijke it was her second marriage since she had married Pieter Johannes Boetes at Achtkarspelen on 30 May 1891. Pieter Boetes, a farmer and merchant, was 69 years old when he married the 29 year old Marijke. They had a daughter Ebeltje Pieters Boetes born on 13 July 1891. Pieter Boetes had died in Rottevalle on 1 May 1897. In addition to his elder half-sister Ebeltje, Hendrik had a younger brother Durk Kloosterman.

Hendrik was brought up as a young boy in the village of Rottevalle which is a small farming village. Although he began his schooling in Rottevalle, he completed his studies at a high school in The Hague.

After graduating from the high school in 1918, Kloosterman entered the University of Leiden. Here he studied mathematics and graduated with a bachelor's degree in 1919, after only one year of study, and a Master's degree in 1922. The education he received in the mathematics department was a good one, his lecturers being J C Kluyver and W van der Woude, but the department did not have stars of the quality that worked in Leiden on mathematical physics. Hendrik Lorentz, although officially retired from 1912, continued to lecture at Leiden and his successor Paul Ehrenfest was an outstanding theoretical physicist whose friends Niels Bohr and Albert Einstein were frequent visitors to Leiden. Of Kloosterman's mathematics lecturers, Kluyver and van der Woude, it was mainly the former that had a strong influence on the young student.

Kloosterman continued his studies at Leiden with J C Kluyver as the supervisor of his number theory thesis. Kluyver, who originally undertook research on geometry, had, after his appointment in 1892 as Professor of Mathematical Analysis at Leiden, changed his research efforts to this field, and in a relatively short time brought analysis teaching to a level that had not previously been reached in the Netherlands. We are now accustomed to analysis being presented according to very rigorous rules, but before Kluyver's time, analysis in the Netherlands was still almost at the level of the 18th century. After Kloosterman arrived in Leiden, rigorous analysis had been established there, and in a short time he proved himself a true master in this field.

It was Ehrenfest, however, who looked after Kloosterman as he would his own students, making sure that he had the opportunity to study with the experts on the subject of his thesis. Ehrenfest arranged for Kloosterman to spend some time in Copenhagen working with Harald Bohr, Niels Bohr's brother.

The topic which Kloosterman was working on for his doctorate was concerned with Waring type problems. Kloosterman was examining the number of solutions in integers *x*_{i}, to the equation

*m*=

*a*

_{1}

*x*

_{1}

^{2}+

*a*

_{2}

*x*

_{2}

^{2}+ ... +

*a*

_{s}

*x*

_{s}

^{2}(*)

Kloosterman presented his doctoral thesis to the University of Leiden in 1924. He had managed to find, provided *s* ≥ 5 and the *a*_{n} satisfy suitable congruence conditions, an asymptotic formula for the number of solution to the equation (*). Under these conditions (1) always has a solution for large values of m. However for *s* = 4 his application of the Littlewood-Hardy method failed and Kloosterman noted in his thesis that it is rather strange that this powerful technique fails to show Lagrange's result that every positive integer is the sum of four squares.

The case when *s* = 4 which Kloosterman had failed to solve in his thesis gave him the challenge which he attacked once his PhD had been awarded. His solution of this case appeared in his paper *On the representation of numbers in the form* *ax*^{2} + *by*^{2} + *cz*^{2} + *dt*^{2} which was published in *Acta Mathematica* in 1926. He writes:-

In this paper Kloosterman introduced what are today called 'Kloosterman sums'. These have proved important in many areas of number theory.In the present paper some special cases of the quadratic form

ax^{2}+by^{2}+cz^{2}+dt^{2}

will be treated by[Hardy's method based on a theorem from the theory of modular functions, following a method which is in principle due to Mordell]. For the sake of simplicity we confine ourselves to the form

x^{2}+y^{2}+cz^{2}+ct^{2}.

We shall prove what we shall call "exact formulae" for the number of representations(as opposed to "asymptotic formulae")forc= 3, 5, 6, 7, but the method used is also applicable forc= 1, 2, 4, 8, 16as well as in a great many other special cases of the formax^{2}+by^{2}+cz^{2}+dt^{2}.

After receiving his doctorate, Kloosterman was required to undertake military service which he did during 1924-25. It was not a wasted year mathematically for he continued to undertake research whenever military duties allowed. The award of a Rockefeller Scholarship allowed him to spend 1926-27 at the University of Göttingen and 1927-28 at the University of Hamburg. During his visit to Hamburg, Kloosterman applied his idea of 'Kloosterman sums' to obtain estimates for the Fourier coefficients of modular forms. It was the Göttingen visit, however, which was most significant in his personal life for there he met Margarete Hilda Träger. Margarete had been born in Grosskromsdork to Friedrich Karl Träger and Auguste Karoline Winkler. Hendrik and Margarete Kloosterman were married on 30 August 1932 in Leiden; they had no children.

After this two years of travel, Kloosterman was offered a post at the University of Münster. He accepted this position and remained there for two years before returning to the University of Leiden in 1930 to a position equivalent to that of associate professor. Springer writes in [7]:-

In fact he taught the first and second year calculus courses for students studying mathematics, physics, astronomy, chemistry, and geology. This was all that he was required to teach but he went further by volunteering to give a different advanced course each year. Among these were courses on [5]:-This was mainly a teaching position. Kloosterman turned out to be an exceptional teacher. He was able to expose with great clarity and great economy the essentials of a piece of mathematics, be it elementary or advanced.

In 1941 the University of Leiden closed during the German occupation of The Netherlands. This in fact presented an opportunity to Kloosterman to undertake research since he had no teaching duties. The university remained closed unto 1945 and the outcome of this period was two major publications on the irreducible representations of finite groups.... the theory of groups, on analytic and algebraic number theory, ideal theory, Galois theory, on quadratic forms, on linear operators in Hilbert space, and on mathematical tools in quantum theory.

The group he studied was the special linear group of 2 by 2 matrices over the ring of integers modulo *p*^{n}. Schur had solved the problem for the case *n* = 1, where the matrices are over a prime field, and the case of *n* = 2 had been solved in the 1930s. Kloosterman solved the general case in two papers *The behaviour of general theta functions under the modular group and the characters of binary modular congruence groups* which occupy 130 pages of the *Annals of Mathematics* in 1946.

When the University of Leiden reopened in the middle of 1945, Kloosterman continued to teach an advanced course each year. He taught courses on representation theory, lattice theory, measure theory, modular functions, elliptic functions and on algebraic function fields. Murre attended these courses and tells us in [5] that these:-

He was promoted to professor at the University of Leiden in 1947, a post he retained until his death. He gave his inaugural lecture on 3 May 1947 with the title... were exciting lectures and modern for that period. They were carefully planned and prepared, masterly presented, starting elementarily but advancing steadily to a high level. Kloosterman was convinced of the unity of mathematics and in his lectures he stressed the connections between different areas. To the audience a beautiful panorama unfolded, completely natural and harmonic. In fact, to attend a lecture of Kloosterman, elementary or advanced, was always a pleasure, both for mathematicians and non-mathematicians. Those of us present there will remember his very clear style, his skilful use of the blackboard(always filled at, but never before, the end of the lecture), his fine sense of humour and certainly also his choice of a 'completely arbitrary' number, which invariably was37.

*Value and valuation of mathematics*(Dutch). The lecture was reprinted in 1984 and Dirk Struik reviewed the reprint writing:-

Kloosterman did outstanding work building up the Leiden Mathematical Institute and also in expanding mathematics at Leiden by being able to set up several new chairs in pure and applied mathematics.Different definitions of mathematics are discussed(Chapman, Kempe, Bôcher, C S Peirce, Schröder)and found to express the nature of mathematics in an oversimplified manner. Mathematics has not only a logical, axiomatic and deductive element, but also an aesthetic and intuitive one.(L Kronecker claimed that mathematicians are poets who also provide proofs for their poems. Nor should intuition and economy of thought be forgotten. And Schopenhauer was wrong when he said, "Where counting begins, understanding stops'', as if mathematics were counting(Rechnen). Schopenhauer also criticized Euclid's method without understanding it. The author then deals with certain objections by Goethe and Lenard, and points out that even the most abstract mathematical structures may find or already have their applications: mathematics never breaks every connection with reality.

He attended the International Congress of Mathematicians held in Cambridge, Massachusetts in 1950 and delivered the lecture *The characters of binary modulary congruence groups* which was published in the Proceedings of the Congress. Robert Rankin writes in a review:-

In 1954 the International Congress of Mathematicians was held in Amsterdam and Kloosterman was chairman of a small committee with the task to draw up a report on structure and regulations of the Congress. See THIS LINK.In two earlier papers the author has considered the problem of determining the characters and representations of the binary modulary congruence groups modulo N by deriving the transformation formulae under modular substitutions of certain multiple theta series. By using binary theta series only he succeeded in determining explicitly the greater part of the characters and irreducible representations and conjectured that the remaining irreducible representations might be determined by considering ternary or higher theta series. ... The author has now discovered that the remaining irreducible representations can be found ... Detailed proofs are to appear elsewhere.

In the 1955-1956 academic year, he was in the United States as a visiting professor at the University of Michigan at Ann Arbor, and also spent some time at the University of Notre Dame.

In [7], as well as looking at Kloosterman's contributions, Springer looks at further developments of his techniques. He writes:-

In [5] Murre quotes P Sarnak on the influence of Kloosterman's work:-Although he was not a prolific writer, his work had a significant impact and is still of considerable interest.

In [4] N G de Bruijn writes:-I am happy to comment on Kloosterman and his influence on modern number theory. His paper in the twenties on quaternary quadratic forms is one of the most influential in analytic number theory. He introduces a refinement of the Hardy-Littlewood circle method which has been at the heart of much modern work(Linnik, Selberg, Iwaniec, Hooley, et cetera). It is such an important refinement that some even refer to the circle method as the Hardy-Littlewood-Kloosterman method. This refinement led him to the Kloosterman sums. Besides introducing these basic objects(they are the 'Bessel functions' of finite fields)he derived their basic properties. It was left to Hasse and Weil to give the well known and deep estimate for these sums. Kloosterman sums find their applications in many proofs of results in automorphic forms, L-functions, quadratic forms ... . He has certainly left his mark on modern number theory(arithmetical and analytic).

Let us mention two honours given to Kloosterman. He was elected as a member of the Koninklijke Nederlandse Akademie van Wetenschappen (Royal Dutch Academy of Sciences) in 1950. In 1986 the Department of Mathematics at the University of Leiden founded the Kloosterman Chair. Every year, beginning in 1986, a distinguished mathematician is appointed as a visiting professor at the Leiden Mathematical Institute. It is usually a two month appointment. The first holder of the Kloosterman Chair was Michael Artin and the article [5] is a slightly modified version of a lecture delivered by J P Murre on 7 May 1986 to mark Michael Artin's visit to Leiden as the first Kloosterman Professor.Kloosterman was a profound mathematician who stood in two worlds. Due to the influence of Kluyver, Harald Bohr and Hardy, he had become a master of classical "hard" analysis. He mastered these, usually difficult, analytical techniques in complete perfection. He was able to perform long and difficult calculations with great patience and incomprehensible accuracy without for a moment losing sight of the goal that had been set. But precisely because his technique was so strong, one must admire the fact that he did not succumbed to the temptation to rest on the laurels that can be obtained with it. In Göttingen and Hamburg he came into contact with modern algebra, which, together with set theory and topology, was to permeate the whole of mathematics. He then set himself the goal of helping to promote its influence. It is in these areas, more than in analytical machinery, that his influence on Dutch mathematical life was strongest. His influence on others was, apart from his profundity and versatility, due to the gift he had to convey his thoughts in word and writing. He also liked it. For example, he was never bored with teaching the first-year students the principles of differentiation and integration. In his lecturing technique, he was strict and yet clear, not a word too many or too few, complete but never boring. Short, dry, humorous remarks occasionally created relaxation. His listeners were at the same time spectators, because he managed to clearly display the entire content of an hour's lecture on one board without ever wiping it out.

**Article by:** *J J O'Connor* and *E F Robertson*