**Aleksandr Yakovlevich Khinchin**'s father was an engineer. Khinchin attended the technical high school in Moscow where he became fascinated by mathematics. However mathematics was certainly not his only interest when he was at secondary school for he also had a passionate love of poetry and of the theatre. He completed his secondary education in 1911 and entered the Faculty of Physics and Mathematics of Moscow University in that year.

At university in Moscow Khinchin worked with Luzin and others. He was an outstanding student being particularly interested in the metric theory of functions and before he graduated in 1916 he had already written his first paper on a generalisation of the Denjoy integral. This first paper began a series of publications by Khinchin on properties of functions which are retained after deleting a set of density zero at a given point. He summarised his contributions to this area with the paper *Recherches sur la structure des fonctions mesurables* Ⓣ in *Fundamanta mathematica* in 1927.

After graduating in 1916, Khinchin remained at Moscow University undertaking research for his dissertation which would allow him to become a university teacher. After a couple of years he began teaching in a number of different colleges both in Moscow and Ivanovo. The town of Ivanovo, east of Moscow, was a centre for the textile industry and it plays a surprisingly important part in the development of Russian mathematics with several of the major figures teaching in the town.

Around 1922 Khinchin took up new mathematical interests when he began to study the theory of numbers and probability theory. In the following year he strengthened results of Hardy and Littlewood with his introduction of the iterated logarithm published in *Mathematische Zeitschrift*. With these ideas he also strengthened the law of large numbers due to Borel.

In 1927 Khinchin was appointed as a professor at Moscow University and, in the same year, he published *Basic laws of probability theory*. Between 1932 and 1934 he laid the foundations for the theory of stationary random processes culminating in a major paper in *Mathematische Annalen* in 1934. Khinchin left Moscow in 1935 to spend two years at Saratov University but returned to Moscow University in 1937 to continue his role of building the school of probability theory there in partnership with Kolmogorov and others, including in particular their student Gnedenko. From the 1940s his work changed direction again and this time he became interested in the theory of statistical mechanics. In the last few years of his life his interests turned to developing Shannon's ideas on information theory.

We shall look at some of Khinchin's major publications and in this way get a feel for the large number of important contributions he made in a remarkably large range of topics. Some of these publications we have already mentioned in the brief description of his career which we gave above.

Khinchin first published the book *Continued Fractions* in 1936 with a second edition being published in 1949. The book consists of three chapters, the first two of which present the classical theory of continued fractions. The third chapter, the longest and most important, contains an account of Khinchin's own contributions to the topic of the metrical theory of Diophantine approximations. Another contribution by Khinchin to number theory is the short book *Three pearls of number theory* which appeared in an English translation in 1952.

The book *Eight lectures on mathematical analysis* by Khinchin ran to several editions. It was first published in 1943 and the eight lectures it contains are: Continuum; Limits; Functions; Series; Derivative; Integral; Series expansions of functions; and Differential equations. The book was designed to be used to supplement a standard course on the calculus and gives a careful treatment of some of the basic notions of mathematical analysis. Ivanov, reviewing the fourth edition, wrote:-

Khinchin publishedThe presentation is smooth, elegant and interesting and makes very enjoyable reading ...

*Mathematical Principles of Statistical Mechanics*in 1943. It showed how to make classical statistical mechanics into a mathematically rigorous subject, developing a consistent presentation of the topic. In 1951 he extended the work of this 1943 book when he published

*Mathematical foundations of quantum statistics*. This new publication on the topic appeared in a German translation in 1956 and then in an English translation in 1960. The book was written in such a way as to be useful both to mathematicians who wanted to become better acquainted with some applications of analysis to physics, and also to physicists who wanted to understand more about the mathematical foundations for their subject. Topics covered included: local limit theorems for sums of identically distributed random variables; the foundations of quantum mechanics; general principles of quantum statistics; the foundations of the statistics of photons; entropy; and the second law of thermodynamics. The book has been rated as being equal in quality to von Neumann's masterpiece

*Mathematical foundations of quantum mechanics*.

Khinchin's book *Mathematical Foundations of Information Theory*, translated into English from the original Russian in 1957, is important. It consists of English translations of two articles: *The entropy concept in probability theory* and *On the basic theorems of information theory* which were both published earlier in Russian. The second of these articles provides a refinement of Shannon's concepts of the capacity of a noisy channel and the entropy of a source. Khinchin generalised some of Shannon's results in this book which was written in an elementary style yet gave a comprehensive account with full details of all the results.

In [6] Gnedenko, who was a student of Khinchin, lists 151 publications by Khinchin on the mathematical theory of probability (the list is given again in [4]).

Among the many honours which Khinchin received for his work was election to the USSR Academy of Sciences in 1939 and the award of a State Prize for scientific achievements in the following year.

Vere-Jones writes [9]:-

Khinchin was a fascinating figure ..., not least because of his early enthusiasms for poetry and acting, and his links with such figures of the revolution as the poet Mayakovsky and members of the Moscow Arts Theatre.

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

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