Volume 38, No.2 - Summer 1992
Editor of this issue: Antanas Klimas, University of Rochester 
ISSN 0024-5089
Copyright © 1992 LITUANUS Foundation, Inc.


Česlovas Masaitis, Ph.D.

Last year (1991) Jonas Kubilius, professor of the University of Vilnius, celebrated his 70th birthday. An International Conference on Number Theory held in Palanga, Lithuania, was dedicated to celebrate his birthday, Last summer scientists from Austria, Czechoslovakia, France, Great Britain, Hungary, Japan, Soviet Union, and the United States — 62 of them — joined 17 Lithuanian mathematicians at this conference and each presented a paper on a topic in the field of his research. This attention to professor Kubilius is primarily due to his contribution to this field of mathematics. Theory of probability and number theory gave the glad eye to each other already in words of P. Erdos, M. Kac, and other mathematicians. However, Kubilius was the "matchmaker" of these two disciplines and the results of his research led to their bond, that generated a new and interesting family of investigations by mathematicians all over the world. Kubilius' contribution to this field was the principal impetus for transferring the Berkeley International (quadrennial) Conference on Number Theory and Theory of Probability from the United States to Lithuania in 1973.

However, this was, perhaps, not the only motive for dedicating the last conference to his anniversary. Besides his scientific achievements, he also deserves credit for his administrative work as the president of the University of Vilnius. He was the 80th president of this school, which is more than 400 years old, and he worked in this capacity for almost 33 years, while majority of the previous presidents held this office for three years only, with the average five years. Besides this, professor Kubilius guided this University through the darkest period of its history, the period of Communist suppression of freedom not only of speech but also of thought.

Professor Jonas Kubilius was born in the village of Fermos, Eržvilkas county, district of Jurbarkas, Lithuania, on July 27, 1921. He was the oldest of the five brothers in his family. He attended Rudkiškiai grade school and Eržvilkas middle school. Then he continued his education in Raseiniai high school where he made highest grades in his class. His main interest was mathematics and literature. When he was in what is equivalent to ninth grade of the United States schools, he already understood all the high school mathematics and occasionally even helped his teacher explain a new lesson in mathematics to his class. He graduated from the high school in 1940 and then began his studies in mathematics at the University of Vilnius. In 1944, he interrupted his studies and became a teacher of mathematics at the middle school in Eržvilkas. in 1945, he returned to the university to continue his studies. At that time there were no courses in number theory at the University of Vilnius, but as a young student, Kubilius was interested in this subject and he chose it for the topic of his graduation paper. He graduated with summa cum laude in 1946.

After finishing his studies he remained on the faculty in the Department of Physics and Mathematics. In 1948, he went to Leningrad (now Sankt Petersburg) for graduate studies. In 1951, he received his master's (candidate's) degree. His master's thesis was Geometry of Prime Numbers. Then he continued to teach at the University of Vilnius as a senior lecturer. In 1952, in addition to his teaching, he started to work at the Institute of Physics and Technology of the Academy of Sciences as a science fellow and a leader of a group of mathematicians at this Institute. In 1953, he was promoted to the position of assistant professor at the University. He also participated in reorganizing the Institute of Physics and Technology into the Institute of Physics and Mathematics, and in 1956 became an assistant to the director of the Institute and the head of the Division of Mathematics, while still continuing to lecture at the University. He developed interesting and original applications of probabilistic methods to unsolved problems in number theory. In 1958, he received the Ph.D. degree in mathematics from the Mathematical Institute in Moscow for his dissertation on the Foundation of Probabilistic Number Theory. In this dissertation, he developed a new approach to the study of number theory based on probabilistic laws. At the same time, he was promoted to professor at the University of Vilnius and in 1962, elected to membership in the Academy of Sciences.

In addition to his teaching, research, and administrative activities, Kubilius was a patron for the Olympiads of young mathematicians in Lithuania and he prepared several books of problems for these Olympiads. He also organized a mathematics seminar in which not only local mathematicians took part but also many scientists came from abroad to lecture to the seminar at the invitation of the University. He also was president of the Lithuanian Mathematical Society which was organized in 1960.

Besides his academic activities he was also active in many cultural, social and even in some political endeavors and published nearly 1,000 articles on various topics in these areas, including the history of science.

In 1958, he was elected president (rector) of the University of Vilnius. That was a difficult time for this task. A few years after he became the president of the University, Khrushchev was removed from the post of the first secretary of the Communist Party and from his leadership of the Soviet Union. That was the end of the mild and short "thaw" of the Khrushchev era and the beginning of the Brezhnev stagnation. The prior president of the University, J. Bulovas, was removed from his office by the Communists, because he failed to purge the University of "bourgeois ideology, relics of nationalism of dinging to the past, and national isolationism." Teaching at the University in Lithuanian language was intolerable "national isolationism" for the Soviet rulers. At the time, the University of Vilnius was the only university in the Soviet Union where all subjects were taught not in Russian but in the native language of the country. The president who failed to correct this had to be removed! Now a new president was elected. Would he succeed in dinging to this "national isolationism?" He did! He managed to preserve this by his ability to manipulate within the complex bureaucratic system of the Soviet Union and mainly because of his international recognition due to his scientific achievements, proof of which was the bestowal of doctor's degrees by several universities. He did more than just preserve the Lithuanian language in the courses of the University. In spite of the Soviet demand that all the research papers be published in Russian, Kubilius encouraged the faculty members to also write in Lithuanian, English, German, and French. it was not possible to have most of the research papers published in these languages, however, some articles in the Lithuanian Mathematical Collection (Lietuvos Matematikos Rinkinys) journal as well as in other scientific journals edited or published by the University personnel appeared in languages other than the language of the "older brother," i.e. Russian. Also in Vilnius most of the student textbooks were Lithuanian. Professor Kubilius himself pre-pared and published Lithuanian textbooks such as Probability Theory and Mathematical Statistics (Tikimybių teorija ir matematinė statistika) published in 1980 and Function Theory of a Real Variable (Realaus kintamojo funkcijų teorija) published in 1970. He also encouraged other faculty members to write text-books for their courses in Lithuanian.

The Soviet bureaucrats did not succeed in quenching his concern for research and education and for the necessary freedom of thought and speech. Also his regard for the ties between the academic activities and the country, its people, their past and their future helped to preserve the proper academic level of the University and, in spite of the severe Soviet obstruction of freedom of thought, brought it to prominence in the free world. Even with the strict restrictions for the Soviet citizens to travel abroad, the University of Vilnius, under the guidance of its president Kubilius, managed to arrange many visits by the faculty members to various universities in the free world to attend scientific conferences there, to lecture and to broaden their knowledge. Kubilius himself lectured at universities in Austria, Canada, France, Finland, Germany, India, Italy, and the United States.

In 1979 the representatives of many Soviet universities and at least of 15 universities of various countries outside the Soviet Union came to Vilnius to celebrate the quadricentennial anniversary of the University. The words of their short greeting speeches testify to the recognition of the academic level of the University of Vilnius.

In his speech at this celebration, president Kubilius said: "The past is the root that is closely joined to this day and frequently it sprouts with unexpected buds even for the future." These words certainly were a heresy for the Soviet ideology and they expressed the president's different view on the role of our past and of the 400-year-old institution of learning. Indeed, looking at the past according to the meaning of the words he uttered is irreconcilable with the Marxist ideology and the effort to renounce the old world and to create a new one as even the Marxist anthem says. Neither did the rest of the Kubilius' speech at this celebration follow the rules of the Soviet style, since he did not mention Lenin or Marx even once. Not only in festive declarations such as at this celebration of the jubilee of the University did one have to pay homage to Lenin, but also every author of a book was expected to write in his introduction that the ideas of the book directly or indirectly were derived from the "infinite wisdom" of Lenin. If he failed to do this, then the "editors" of the publishing institution added it for him. Thus, this official address by president Kubilius illustrates his attitude toward the role of the university in the nightmare of communism.

Although he was a member of the Community Party (of course, without being a member of this organization he would hardly have been able to guide the University through the jumble of Soviet lies and intimidations), his ideology was quite different from the spiritual degradation inherent in Marxist doctrine, as the preceding remarks indicate. Besides having good reason to reject the insanity of Marxist ideology, Kubilius, perhaps, did so also because of his personal experience that could not induce any affection for the Communist system. In 1948, his younger brother Juozas was sentenced to the concentration camp for 25 years because of his alleged ties with the national resistance movement. His mother and his brother Antanas were also deported to Siberia, the third brother, Bronius, was expelled from the university as a student, because he was unacceptable to the communists.

Another illustration of the difference of Kubilius' views from communist assumptions is the following quotation from his speech at the ceremony of raising the University bell at the early dawn of freedom in 1989: "The bell is a symbol of the Feast of the Resurrection. By what words could one express the unique state of our soul when Easter bells ring early in morning of the spring?" In the same speech his words: "Not all the circumstances of its (the University's) history were favorable for fostering the unhindered Lithuanian national consciousness. However, from the very beginning of the establishment of this Academic institution the effort to resist the coercion of spirit never succumbed" testify to the struggle of the University for freedom of thought under his guidance.

His unwavering dedication to the higher principles, his ability to fend off frequent attacks by the communist party's nomenclature, and — according to the words of some of his friends — also lucky chance events allowed him to guide the University through the most difficult period of its existence for almost 33 years, until he resigned his post early in 1991 to pass the helm of the University to a younger person.

Kubilius did an excellent job in his post as president in a really difficult time. However his 70th birthday is remembered primarily because of his achievement in mathematical research. As it was already mentioned above, his research was in number theory. This is one of the oldest branches of mathematics, and Kubilius created a new blossoming twig of this branch. The German mathematician of the 19th century, Carl Friedrich Gauss, called this branch Queen of Mathematics.

The main interest of this branch is in the set of prime numbers, i.e. integers that are divisible only by 1 and each by itself. Examples of primes are: 2, 3, 5, 7, 11 etc. The main question in the number theory is how many primes are there, i.e. how many primes not exceeding a certain preselected integer and how many of them there are all together. The last question was answered by the Greek mathematician Euclid in the fourth century B.C. His solution was very simple: Assume that there are finitely many primes. Compute the product of all and add 1 to it. You get a number that is greater than any one of the primes used to compute the product and also not divisible by any of them. Thus this number is either a prime or else it is divisible by a prime not included in its computation. Thus, in both cases we have a contradiction resulting from the assumption that there are only finitely many different primes. Therefore the right conclusion is that there are infinitely many primes.

The other question: how many primes are there each less than preselected number is much more difficult. We note that there are four primes not exceeding 10, namely 2,3, 5, and 7. If the frequency of primes were the same in every sector of integers, we would have 40 primes not exceeding 100 and 400 primes not greater than 1,000. However, there are only 25 primes not exceeding 100, only 168 primes less than 1,000, and only 135, not 168 primes between 1,000 and 2,000. Then, how many primes are there in a given interval, say between quadrillion and quintillion? Kubilius derived a formula to compute a much more accurate estimate of this number, than similar formulas derived earlier.

The following is another problem in number theory solved by Kubilius. Some of the primes can be expressed as the sum of the square of two integers, and others cannot be expressed in this fashion. For instance, 5 is the sum of the squares of 2 and 1,13 is the sum of the squares of 3 and 2, i.e. the sum of 9 and 4. The primes 7, 11, 19, 23 are not sums of squares of any two integers. The question then is: How many primes are there that can be expressed this way? In his paper published in 1951, J. Kubilius proved that there are infinitely many such primes.

These are examples of simple problems solved by Kubilius. However, the solutions were not simple in any way. Many mathematicians in the past tried to solve these problems and did not succeed or succeeded only partially. Kubilius solved them and many other problems in number theory by applying the method of probability theory. For this purpose, he defines certain probability spaces over the set of integers and uses the property of the corresponding probability functions to solve problems of number theory. Most of his solutions are obtained by analyzing statistical properties of so-called additive and multiplicative arithmetic functions. This analysis is very sophisticated and it looks very complex even to mathematicians working in this field of mathematics, as can be illustrated by some remarks in the reviews of Kubilius' paper in Mathematical Reviews. For instance, in 1953, A. E. Ingham writes in his review of Kubilius' paper: "The results are too elaborate to summarize in detail." Thus, they are too elaborate even for a review meant for mathematicians! In another review of Kubilius' paper, J. Galombos states: "To obtain several terms of his expansion, he has to carry out careful estimation of complicated integrals, a skill that he has developed for this kind of situation in a series of his papers." In his review of still another Kubilius' paper, W. J. Le Veque calls his analysis "inherently complicated." For this reason, in order to avoid a highly technical mathematical vocabulary, only two simple results of Kubilius' research are mentioned above and even these without any indication how he solved them.

This complicated research by Kubilius developed a new branch of mathematics of which Soviet academician Y. V. Linnik commented: "Jonas Kubilius' research is a great contribution to science where two domains of mathematics — probability theory and number theory — intersect. Only some considerations and isolated facts existed in this field before. Now he has shaped a comprehensive and far-reaching theory. The essential parallelism of number theory and that of probability has been established. This can be considered as having even a philosophical significance."

Kubilius compiled his created axioms and methods of analysis in a monograph: Probabilistic Methods in the Theory of Numbers. Its first Russian edition was published in 1959 and its second revised edition appeared in 1962. This was translated into English and published in the United States in 1964. Additional English publications of this monograph appeared in 1968 and 1978. In 1967, The Bulletin de la Societe Mathématique de Belgique printed the following comment on this monograph: "The work is an epoch-making event in its originality and in the effectiveness of the principle used." In the introduction to his two volume monograph Probabilistic Number Theory, P. D. T. Elliot writes: "Many of the results in Kubilius' monograph will be contained amongst the theorems of the present book. Some, such as the study of truncated additive functions for their awn sake, will not. Besides lack of space, we have at present little to add to such a topic. Thus Kubilius' book remains a useful reference."

Prof. Kubilius' contribution to mathematical research is not only in the field of number theory. He is also the founder of the school of theory of probability in Lithuania. His students such as academicians B. Grigelionis, V. Statulevičius, E. Vilkas and others are active in this field and have gained international recognition for their research. In the History of University of Vilnius 1940-1979 professors A. Bikelis and E. Monstavičius write: "]. Kubilius is the leader of mathematics in Lithuania. He is the founder of the journal Lithuanian Mathematical Collection (Lietuvos Matematikos Rinkinys), originator of the sector of mathematics in the Lithuanian Academy of Sciences. This sector developed into the Institute of Mathematics and Cybernetics. He is also the initiator and fosterer of new branches of mathematics in Lithuania, the organizer of national and international science conferences." The journal Lithuanian Mathematical Collection (Lietuvos Matematikos Rinkinys), was initiated by Kubilius in 1961. Most of the papers in this publication are contributed by Lithuanian mathematicians. Kubilius himself has published a number of his research papers in this journal. It is a valuable source of new results of mathematical research as is shown by the fact that it is translated into English and reprinted in the United States. Libraries of many universities and research institutions in the United States subscribe to it.

These are some of the reasons why in 1991 the International Conference on Number Theory was dedicated to his 70th birthday and why his countrymen in Lithuania and in foreign countries wish him multos annos!