**John Knopfmacher**was the son of Walter and Ruth Knopfmacher. Walter Knopfmacher was born on Berlin on 6 August 1899 and attended the Koellnisches Gymnasium and the Friedrichs-Werdersche Oberrealschule in Berlin. After serving in World War I, he studied at the technical universities in Berlin and Breslau and graduated in March 1922 in Breslau with a Diploma in electronic engineering. He worked for the General Electricity Society, Berlin, then with the N V Molybdenum Company Amsterdam in Reutte, Tirol. He emigrated to South Africa and died in 1970.

John's son, Arnold Knopfmacher, writes about his father's childhood [1]:-

John attended primary school at Yeoville Boys' School, and high school at Athlone Boys' High. Jack Rose writes [3]:-As a high school student John's marks were not exceptional, a fact that perhaps can be linked to an unhappy home life. John's father was himself a holder of a professorship and Ph.D.(in Engineering)but John received little encouragement from his parents. He related for example, how while occupied with a painting he was working on, it was suggested that he would be better employed in painting the house. He did however become reconciled with his mother towards the end of her life, which allowed us children to know a grandparent for the first time in our lives.

After graduating from High School, Knopfmacher entered the University of the Witwatersrand in Johannesburg to study accountancy. He was not happy with the accountancy course, however, and after one year studying that topic, he changed to take mathematics courses. Arnold Knopfmacher writes [1]:-We had been boyhood friends for a number of years ... We stayed near each other in South Africa, and were in the same school classes for a number of years. I remember that in primary school he expressed himself well in composition, and had lots of interesting ideas for a child of11years. In high school we played in the same musical instruments band - he played the euphonium and I played the tuba. The only reason that we volunteered for such big, heavy instruments was that the smaller ones - like trumpets and clarinets - were no longer available. After some time we both left the band - and nobody asked us to come back. Although John was a good student, he was not brilliant in high school.

The publication,He related to me that as only a first year student, he became inspired by the famous problem of odd perfect numbers and derived what he believed to be a proof that none existed. One of his lecturers realised that while not a proof of this, it was in fact a new proof of the formula that describes all even perfect numbers, and this result became his first publication in a mathematics journal.

*Note on Perfect Numbers*, appeared in volume 44 of the

*Mathematical Gazette*in 1960. By the time Knopfmacher was in his third year of study at the University of the Witwatersrand his lecturers were so impressed with his work that he was asked to give some lectures to first year students. He was awarded a B.Sc. in 1958, followed by First Class Honours in Applied Mathematics in 1959, and, after one more year of study, First Class Honours in Mathematics in 1960.

He married Rose Hendler on 6 January 1959; they had three children, Arnold Knopfmacher, Nadine Knopfmacher and Kevin Knopfmacher.

He was awarded the Rusterholz Memorial Scholarship for his M.Sc. which he received in 1961 from the University of the Witwatersrand for his thesis *A quantum-mechanical type of mathematical system*. He was awarded the J H Hofmeyr Postgraduate Scholarship to enable him to study for a Ph.D. in England and he travelled to the University of Manchester to undertake the necessary research. His wife Rose and their five month old son Arnold travelled with him to Manchester. At Manchester his thesis advisor was J Frank Adams.

In May 1962 he submitted the paper *Universal envelopes for non-associative algebras* for publication in the *Quarterly Journal of Mathematics* and it appeared in 1962. In it he gives the following acknowledgement:-

We note that Michael George Barratt (1927-2015) studied at Brasenose College, University of Oxford, before being appointed to a junior lectureship at the University of Oxford and as a fellow of Magdalen College in 1950. After spending the year 1956-57 in the United States at Princeton and Chicago University he was appointed to a Senior Lectureship at the University of Manchester in 1959. His main mathematical interests were in algebraic topology.... the author would like to express his deep gratitude to Dr M G Barratt for many extremely helpful comments and suggestions on these topics.

Arnold Knopfmacher writes [1]:-

[Knopfmacher was awarded a Ph.D. in 1965 for his thesisJohn Knopfmacher]spoke little of this period, but I gather that he largely chose his own area of study and completed the work with little supervision. My impression is that the academic culture and traditions of England struck a chord with him and that he returned to South Africa, now with the addition of a baby daughter Nadine, mainly to please Rose who greatly missed her family here.

*Extensions in Varieties of Groups and Algebras*. Before the award of his doctorate, he submitted the article

*On Chern classes of representations of finite groups*to the

*Bulletin*of the American Mathematical Society. He gives the following acknowledgement in the paper:-

Doron Shaul Lubinsky makes the following comment about Knopfmacher's thesis [6]:-The purpose of this note is to announce the proof of a conjecture of J F Adams ...; the main idea of the proof was suggested to me by Professor J F Adams, and is believed to emanate essentially from Professor Atiyah. I would like to thank Professor J F Adams sincerely for his help, and to acknowledge the helpfulness of Professor Atiyah and Professor M G Barratt.

After returning to South Africa, Knopfmacher was appointed as a lecturer at the University of the Witwatersrand, Johannesburg [1]:-His thesis received instant recognition. Its main results were published in the prestigious 'Acta Mathematica' of Sweden. This was followed by a series of articles on several aspects of algebra, including non-associative algebra, finite groups, Lie algebras, homology, and finite topological spaces, in journals such as the 'Bulletin' of the American Mathematical Society, and the 'Oxford Quarterly Journal of Mathematics'.

Knopfmacher was offered promotion but was reluctant to accept it. He was, however, promoted to Senior Lecturer in 1966, and to Reader and Associate Professor in 1971. His research was innovative and introduced important new ideas as Doron Lubinsky explains in [6]:-At the time he joined Wits Mathematics Department as a lecturer, only he and the head of Department, held Ph.D. degrees. From this time in1965until the early eighties he found himself as regards research, in a position of both geographical and mathematical isolation. Despite this, his inner drive never wavered and perhaps his most significant works, in abstract analytic number theory, a subject he invented and developed himself, appeared during this time.

In 1975 he published the bookIt was in1970that his first publication appeared on abstract analytic number theory. He had realized that the huge body of results of classical number theory - such as the prime number theorem - that treat problems involving prime numbers in the positive integers, could be placed in a vastly more general and abstract setting. This allowed simultaneous treatment of a very broad range of enumeration problems, varying from the most concrete to the most abstract, from the continuous and infinite to the discrete and finite. For this purpose, he invented and developed the basic notions of arithmetical semigroups and formations. In a profound series of papers, starting with an announcement in the 'Bulletin' of the American Mathematical Society, and developed in six successive papers in the 'Journal für die Reine und Angewandte Mathematik' from1972to1975, he laid the foundations for the theory, and obtained many of its most significant results. The power and range of his results, their ability to unify diverse phenomena, and his development of a totally new area of number theory attracted worldwide attention.

*Abstract analytic number theory*. W Narkiewicz writes in a review:-

In 1978 Knopfmacher was offered a full Professorship of Mathematics at the University of the Witwatersrand. After initially refusing the position on the grounds that he would become too involved in administration, he eventually accepted when it became clear that there was no other suitable person for the position. He gave his Inaugural Lecture,This book is devoted to the study of arithmetical properties of free commutative semigroups with unit element, generated by an at most countable set of free generators and equipped with a real-valued norm which is multiplicative, exceeds unity at free generators and, moreover, is such that for every real x the numberN(x)of elements a with norm|a| ≤xis finite. Such a semigroup is called an arithmetical semigroup. ... The book has an interesting appendix in which the author poses several unsolved questions and gives many interesting proposals for the further development of the theory exposed in the book. The book is well written and the bibliography excellent.

*Some Aspects of the Theory of Prime Numbers*, in 1979. In the same year he published the book

*Analytic arithmetic of algebraic function fields*. W Narkiewicz writes in a review:-

In 1984 Knopfmacher agreed that he would accept the position as Head of the Mathematics Department and the City of Johannesburg Chair. He continued as Head of Department until 1994.The author considers arithmetical semigroups ... There are many natural examples of such semigroups, e.g., Galois polynomial rings, the set of finite modules over such a ring, etc. The author obtains many results concerning the analogues of various theorems in analytic number theory in this general setting. These include analogues of the prime number theorem, sum estimates for arithmetical functions, maximal orders for them, and error estimates in various asymptotic formulas. This book is a companion to another book of the author[Abstract analytic number theory(1975)]in which another type of arithmetic semigroup is considered.

Knopfmacher's son Arnold Knopfmacher studied mathematics at the University of the Witwatersrand and was awarded a Ph.D. in 1985 for his thesis *Linear operators and Christoffel functions associated with orthogonal polynomials*. His thesis advisor had been Doron Lubinsky. Beginning in 1987 John and Arnold published joint papers, their first being *A new infinite product representation for real numbers*. Here is the Abstract of that paper:-

The father and son team produced around 35 joint papers, the last in 2000 being a three author paper, the other author being George E Andrews. This paper,We introduce a new algorithm that leads to a representation for any real number greater than one as an infinite product of rational numbers. Just as we can regard the Cantor product as being a product analogue of the series of Sylvester, this new product is analogous to the classical Engel representation for real numbers. The growth conditions satisfied by the digits in the product are likewise shown to correspond to those required for the Engel series. The representation for certain types of rational numbers via this algorithm is also considered.

*Engel expansions and the Rogers-Ramanujan identities*, has the following note attached:-

As Head of Mathematics at the University of the Witwatersrand, Knopfmacher made efforts to increase the research profile of the department. In 1992 he founded the Centre for Applicable Analysis and Number Theory at the University to provide a focus for research. This brought research visitors into the department and provided a very positive boost. Richard Warlimont, who worked at the Universität Regensburg, Germany, collaborated with both John Knopfmacher and his son Arnold Knopfmacher. Warlimont wrote after Knopfmacher's death [7]:-It is with great sadness that we note that John Knopfmacher passed away on May29,1999at age62. He will live on through his research, in particular through his three books on Abstract Analytic Number Theory.

In September 1991 Knopfmacher and his wife Rose were divorced and on 23 October 1991 he married Beverley Woolf. In 1997 Knopfmacher retired from the University of the Witwatersrand and moved to Melbourne, Australia. He retained his link with the Centre for Applicable Analysis and Number Theory at the University of the Witwatersrand as an Honorary Professorial Research Fellow. Arnold Knopfmacher writes [1]:-John Knopfmacher, Johannesburg, has been a concept in the mathematical community which stood for excellence. Therefore I felt so much honoured when in1989he invited me to Wits and still more so when I became his friend. Ever since we worked together and we were complementary in such a favourable way: he contributed the ideas for the chosen subject and outlines for proofs etc. while I then was engaged in the details.

His 1975 bookIn July1997John moved to Melbourne, where he and his second wife Bev, hoped to find a better quality of life. He loved the beautiful coastal scenery and the tranquillity of his new environment but at the same time greatly missed Wits and his three children. He was an honorary visitor at Melbourne University but had hoped to find a proper mathematical position, even at a junior level, but was unsuccessful in this. This unfortunately left him feeling disappointed and unsure of the value of his mathematical works.

*Abstract Analytic Number Theory*became a classic and, in 1990, it was republished in paperback by

*Dover*. After he went to Australia, he worked with Wen-Bin Zhang on an updated and extended edition of his 1979 book

*Analytic arithmetic of algebraic function fields*which the authors entitled

*Number theory arising from finite fields: Analytic and probabilistic theory*. It was published in 2001, two years after Knopfmacher's death. Wolfgang Schwartz writes in a review:-

In 1999 Knopfmacher went to Graz, Austria, as a Visiting Professor. He died in the Mathematics Department of the University of Graz on Saturday 29 May 1999, around 1 p.m. Robert F Tichy writes [7]:-In1979J Knopfmacher's monograph 'Analytic arithmetic of algebraic function fields' appeared in print. Since then, according to the preface of this new book, "some remarkable advances have been made in many directions in this area, particularly regarding abstract prime number theorems, the theory of additive formations, mean-value theorems for multiplicative functions, the probabilistic theory of distribution of values of additive functions, as well as the theory of factorization of polynomials over finite fields, Ramanujan expansions, etc. We now have a rather mature and rich theory which is well developed, and it is therefore time to give a readable account of the theory in a new book." So this new monograph is a largely increased, totally new version of the above-cited monograph from1979, and many of the new results are due to the second author, Zhang(and to A Hildebrand, K-H Indlekofer, A Knopfmacher, J Knopfmacher, E Manstavičius, G Tenenbaum, R Warlimont, ...).

Among the honours Knopfmacher received, we mention the medal for lifetime achievement from the South African Mathematical Society in 1995, and his election to a fellowship of the Royal Society of South Africa in 1991. He served the South African Mathematical Society in his capacity as an editor of its journal,When I left Graz, one week ago, I talked to him on his plans for his series of lectures here in Graz ... John worked many hours every day at the department, and he also visited various places in Graz as a tourist. The students liked his courses very much. Obviously, he prepared further courses on Saturday in his office room when he died.

*Quaestiones Mathematicae*for many years. He was a member of several mathematical societies including the American Mathematical Society, the Edinburgh Mathematical Society and the London Mathematical Society. In September 1999 the Council of the University of the Witwatersrand agreed to rename the Centre Knopfmacher founded in 1992 in his honour as the John Knopfmacher Centre for Applicable Analysis and Number Theory.

As to his hobbies and interests other than mathematics and his family, we mention his love of comedies in both film and radio. He was an avid reader of *Punch* magazine in the 1960s and 1970s. His other loves were painting and music.

Let us quote from the tributes in [7]:-

From Peter Sarnak:

[From Kathleen Ann Driver:John Knopfmacher]played a big role in putting South Africa on the international mathematical research scene through his many papers and books. It is a great loss to Mathematics and of course to all his family, students and friends. His courses that I took as an undergraduate(in algebra and number theory)opened my eyes to fascinating problems in this area and have continued to be main interest since then. I certainly owe a lot to him for this and sadly I never properly thanked him.

Finally we quote from a tribute by his wife [8]:-John was an extremely good friend to me over the years. When I embarked on a Ph.D., he showed the strongest possible support, by reducing my teaching assignments from4courses to2and even, at one point, reduced my duties further. It is clear that he played a big part in enabling me to become active in research and I will always be indebted to him for his unwavering support and friendship. When John stepped down as head, I approached John Hartney and we went to John's office to ask him whether he indeed did not want to be considered again. He seemed pleased by our approach - there are few people who recognise just how difficult the job is as Head of Department.

John was such a quiet, modest man. He had a wonderful sense of humour. ... He loved his family more than anything else in the world, and he loved Mathematics as he loved his family. He died doing the things he loved most. He saw a gentle poetry in Mathematics, I could never imagine what it could be. If you observed him and he was not aware you were watching him, his eyes had a veiled quality as though this was his secret love. The tributes that have flooded in have confirmed what a great man and great mathematician he was. However not everyone knew what a wonderful family man he was. He deeply loved his children, and stepchildren and he was undoubtedly my soul mate. When I think of John I think of everything that is good and pure. A man who would not be compromised, a man of integrity.

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