In the mid-1980s, Anatolii AlekseevichKaratsubamade, for himself, a chronological collection of xerocopies of his papers in four volumes provided with hand-written comments.2 The present paper is compiled from fragments of these records (in fact, a diary): some of Karatsuba’s comments are cited completely, while some others, only partially (compilers’ explanations are given in footnotes; the omitted lines are replaced by asterisks and the omitted words are replaced by dots). In 1990–2000, Karatsuba supplemented some comments with brief remarks on the further development of their topics (here, these more recent remarks are set in oblique type). From 1954 to 1959, Anatolii Karatsuba studied at the Mechanics and Mathematics Faculty, Lomonosov Moscow State University. His first scientific advisor was Andrei Nikolaevich Kolmogorov; under his supervision, Karatsuba wrote his third-year term paper entitled “On variations of functions of two variables” and fourth-year term paper “Experiments with automata.” On “Experiments with automata,” Karatsuba’s first published paper was based, with which the present comments begin. Karatsuba often recalled his student years, his teachers and advisors, professors at Moscow State University. He remembered ski jaunts organized for students by Kolmogorov, who kept 20 pairs of skis and 20 pairs of ski boots at his dacha for this purpose, as well as more than 20 pairs of slippers and cups for subsequent tea with students. Karatsuba’s daily record of November 19, 1985, reads: “Listening to Kathleen Ferrier3 (Schumann in German), remembered Komarovka4, Pavel Sergeevich5, and Andrei Nikolaevich6, their musical concerts after skis. Extraordinary atmosphere, incomparable with anything (and, apparently, will never recur). Different time, different attitude to science. Rare pragmatism and careerism.” Having chosen number theory as the main direction of his research, Karatsuba received PhD (Candidate of Science) degree in 1962; his thesis was entitled Rational Trigonometric Sums of a Special Form and Their Applications (the thesis advisor was Nikolai Mikhailovich Korobov). In 1966, Karatsuba presented his Doctor of Science thesis entitled Method of Trigonometric Sums and Mean-Value Theorems and took the position of researcher at Steklov Mathematical Institute, Russian Academy of Sciences (MIAN), which was headed by Ivan Matveevich Vinogradov at that time. Karatsuba always admired the scientific achievements of Vinogradov. In [1], he wrote about Vinogradov: “He solved Goldbach’s problem, which has resisted all efforts of scientists during centuries, and Vinogradov’s method of trigonometric sums has drastically changed the character of analytic number theory and opened a new era in this classical area of mathematics”. In the same paper, he wrote: “Using a few mathematical notions and facts, he penetrates the essence of problems so deeply that this enables him both to create an apparatus necessary for solving the problem and to solve it.”
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