Komaravolu Chandrasekharan (1920-2017)


Komaravolu Chandrasekharan, a number theorist died earlier this year. He was 97. Chandrasekharan was a student of C.R. Rao (who is known Cramer-Rao bound of statistics). Komaravolu Chandrasekharan got his PhD in 1945 from Madras University. He was a specialist in analytic number theory.

Here is a summary of his life from the Tata institute:

Komaravolu Chandrasekharan was born on 21 November 1920 in Machilipatnam in modern-day Andhra Pradesh. He attended a District Board School in Guntur District, Andhra Pradesh, and then the High School at Bapatla, also in Guntur. He then obtained his M.A in Mathematics from the Presidency College, Chennai and was a Research Scholar in the Department of Mathematics of the University of Madras during the years 1940-1943. During 1943-46 he was a part-time Lecturer at the Presidency College and obtained his Ph.D during this time under Ananda Rau, who was with Ramanujan in Cambridge. Chandrasekharan then went to the Institute for Advanced Study, Princeton, USA. In 1949, while he was in Princeton, he was invited by Homi Bhabha to join the School of Mathematics of the Tata Institute of Fundamental Research. As an extraordinarily gifted organizer and administrator of science, he transformed the fledgling School of Mathematics of TIFR into a center of excellence respected all over the world. He initiated a very successful program of recruitment and training of Research Scholars at TIFR. The program continues to this day along the same lines that he set down. He used his contacts with the leading mathematicians of the world, persuading many of them (like L. Schwartz, a Fields medalist, and C.L. Siegel) to visit TIFR and deliver courses over periods of two months and more. The lecture notes prepared out of these lectures and published by TIFR enjoy a great reputation in the world mathematics community to this day. During 1955-61, he was a member of the Executive Committee of the International Mathematical Union. He served as the Secretary of IMU during 1961-66 and as President during the years 1971-74. His initiatives over a long period of 24 years on this Committee were numerous and valued greatly. He served as the Vice President of the International Council of Scientific Unions during 1963-66 and as its Secretary General during 1966-70. He was a member of the Scientific Advisory Committee to the Cabinet, Government of India during 1961-66. He was awarded the Padma Shri in 1959, Shanti Swarup Bhatnagar Award in 1963 and the Ramanujan Medal in 1966. He was responsible for the IMU sponsoring the International Mathematical Colloquium held every 4 years at the Tata Institute starting 1956. In 1957, on his initiative, TIFR published the Notebooks of Srinivasa Ramanujan. In the fifties, Chandrasekharan held the editorship of the Journal of the Indian Mathematical Society. Thanks to his abilities at persuading some of the great names in the field to publish there, several great papers appeared in the journal during this period. In 1965 he left the TIFR and moved to the Eidgenössische Technische Hochschule, in Zürich. He worked in the fields of number theory and summability.
I had been fortunate to take an introductory number theory course from him as a sophomore student at ETH. As custom, there was no book for this course. The lectures were clear and sufficient. I remember the lectures vividly. "Chandra", (that is how we students would call him), obviously had a lot of fun giving the lectures. He could chuckle over even seemingly dry proofs and make us fall in love with the subject. He also made sure that we appreciated even seemingly simple facts in number theory and saw them in a historical context, where first attempts were often incomplete.
Below are my handwritten lecture notes (34 pages). These notes are not verbatim. I had at that time been fanatic in abbreviate as much as possible. Definitions were color coded in red, results in yellow, examples in green. I would used logical quantors to abbreviate even if this was not in the words of the lecturer. But the notes can gives an idea about a excellent structure and taste Chandra had: Chapter I: Number theory in Z, Chapter II Congruences, Chapter III: Diophantine approximations, Chapter IV: quadratic residues, Chapter V: Geometry of numbers. The notes are actually quite well covered by Chandra's book on analytic number theory.
Lecture Notes
I later took also a pro-seminar with him. Taking such seminars was mandatory and tough! I obtained the task to present a density theorem of George Polya on Dirichlet series. The assistant Albert Stadler (a PhD student of Chandra) had been in charge to make sure that we came prepared to the classes. Preparing for such seminars was a lot of work. Reading the article, putting together the necessary background and bringing down the proof to a 2 hour lecture was not easy. Below are my first preparation notes, directly assembled from the original work of Polya and a book of Levinson. In a second pass, I had condensed the proofs onto index cards for the presentation. Unfortunately, I don't have those preparation cards any more.
Seminar preparation notes and notes from that seminar.
Here are some books authored by Chandra. Each of them is a masterpiece. It shows how much Chandra cared about teaching.
Here are some entries from the Mathematical Reviews which gives an idea about his mathematical work:
  • MR2858258 Chandrasekharan, K. A course on integration theory. Reprint of the 1996 edition. Texts and Readings in Mathematics, 8. Hindustan Book Agency, New Delhi, 2011. viii+118 pp. ISBN: 978-93-80250-19-9 28-01
  • MR2857782 Chandrasekharan, K. A course on topological groups. Reprint of the 1996 original. Texts and Readings in Mathematics, 9. Hindustan Book Agency, New Delhi, 2011. viii+117 pp. ISBN: 978-93-80250-20-5 22Dxx (43A65)
  • MR1640163 Chandrasekharan, K. The autobiography of Laurent Schwartz: review of Laurent Schwartz, un mathematicien aux prises avec le siècle [Editions Odile Jacob, Paris, 1997]. Notices Amer. Math. Soc. 45 (1998), no. 9, 1141-1147. 01A70
  • MR1662921 Chandrasekharan, K. A course on integration theory. Texts and Readings in Mathematics, 8. Hindustan Book Agency, New Delhi, 1996. viii+118 pp. ISBN: 81-85931-09-7 (Reviewer: R. G. Bartle) 28-01
  • MR1661135 Chandrasekharan, K. A course on topological groups. Texts and Readings in Mathematics, 9. Hindustan Book Agency, New Delhi, 1996. viii+117 pp. ISBN: 81-85931-10-0 (Reviewer: Joseph Max Rosenblatt) 22Dxx (43A05 43A65)
  • MR0808396 Chandrasekharan, K. Elliptic functions. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 281. Springer-Verlag, Berlin, 1985. xi+189 pp. ISBN: 3-540-15295-4 (Reviewer: Marvin I. Knopp) 11F11 (11-02 33A25)
  • MR0697150 Chandrasekharan, K.; Good, A. On the number of integral ideals in Galois extensions. Monatsh. Math. 95 (1983), no. 2, 99-109. (Reviewer: Robert Perlis) 12A70 (10H26 12A55)
  • MR0735669 Chandrasekharan, K. Prof. Tirukannapuram Vijayaraghavan (1902-1955). Bull. Math. Assoc. India 14 (1982), no. 1-4, 8-12. 01A70
  • MR0552675 Chandrasekharan, K.; Narasimhan, Raghavan Exponential sums associated with algebraic number fields. Comment. Math. Helv. 54 (1979), no. 4, 523-561. (Reviewer: S. A. Stepanov) 10G10
  • MR0434978 Chandrasekharan, K.; Narasimhan, Raghavan Exponential sums associated with the Dedekind zeta-function. Comment. Math. Helv. 52 (1977), no. 1, 49-87. (Reviewer: Matti Jutila) 10G05 (10H10)
  • MR0414282 Chandrasekharan, K. The work of Enrico Bombieri. Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, pp. 3-10. Canad. Math. Congress, Montreal, Que., 1975. 01A70 (10-03)
  • MR0369277 Chandrasekharan, K. Exponential sums in the development of number theory. Proceedings of the International Conference on Number Theory (Moscow, 1971). Trudy Mat. Inst. Steklov. 132 (1973), 7-26, 264. (Reviewer: A. A. Karacuba) 10G05
  • MR0332678 Chandrasekharan, K.; Joris, H. Dirichlet series with functional equations and related arithmetical identities. Collection of articles dedicated to Carl Ludwig Siegel on the occasion of his seventy-fifth birthday, II. Acta Arith. 24 (1973), 165-191. (Reviewer: J. Knopfmacher) 10H05
  • MR0277490 Chandrasekharan, K. Arithmetical functions. Die Grundlehren der mathematischen Wissenschaften, Band 167 Springer-Verlag, New York-Berlin 1970 xi+231 pp. (Reviewer: L. Kuipers) 10.43
  • MR0249372 Chandrasekharan, K.; Narasimhan, Raghavan An approximate reciprocity formula for some exponential sums. Comment. Math. Helv. 43 1968 296-310. (Reviewer: B. Berndt) 10.41
  • MR0249348 Chandrasekharan, K. Introduction to analytic number theory. Die Grundlehren der mathematischen Wissenschaften, Band 148 Springer-Verlag New York Inc., New York 1968 viii+140 pp. (Reviewer: L. Kuipers) 10.00
  • MR0225751 Chandrasekharan, K.; Narasimhan, Raghavan Zeta-functions of ideal classes in quadratic fields and their zeros on the critical line. Comment. Math. Helv. 43 1968 18-30. (Reviewer: T. Kubota) 10.66
  • MR0211967 Chandrasekharan, K.; Narasimhan, Raghavan On lattice-points in a random sphere. Bull. Amer. Math. Soc. 73 1967 68-71. (Reviewer: F. Supnick) 10.46
  • MR0213282 Chandrasekharan, K. Einführung in die analytische Zahlentheorie. (German) Lecture Notes in Mathematics, No. 29 Springer-Verlag, Berlin-New York 1966 iv+199 pp. (not consecutively paged). (Reviewer: A. E. Ingham) 10.00
  • MR0160765 Chandrasekharan, K.; Narasimhan, Raghavan On the mean value of the error term for a class of arithmetical functions. Acta Math. 112 1964 41-67. (Reviewer: S. L. Segal) 10.43
  • MR0153643 Chandrasekharan, K.; Narasimhan, Raghavan The approximate functional equation for a class of zeta-functions. Math. Ann. 152 1963 30-64. (Reviewer: S. Chowla) 10.41
  • MR0150111 Chandrasekharan, K.; Narasimhan, Raghavan The average order of arithmetical functions, and the approximate functional equation for a class of zeta-functions. Rend. Mat. e Appl. (5) 21 1962 354-363. (Reviewer: A. E. Ingham) 10.41 (10.43)
  • MR0140491 Chandrasekharan, K.; Narasimhan, Raghavan Functional equations with multiple gamma factors and the average order of arithmetical functions. Ann. of Math. (2) 76 1962 93-136. (Reviewer: A. E. Ingham) 10.43 (10.41)
  • MR0171761 Chandrasekharan, K.; Narasimhan, Raghavan Hecke's functional equation and arithmetical identities. Ann. of Math. (2) 74 1961 1-23. (Reviewer: B. V. Levin) 10.43 (10.45)
  • MR0123543 Chandrasekharan, K.; Narasimhan, Raghavan On Hecke's functional equation. Bull. Amer. Math. Soc. 67 1961 182-185. (Reviewer: S. Bochner) 10.41 More links Review PDF Clipboard Journal Article
  • MR0126423 Chandrasekharan, K.; Narasimhan, Raghavan Hecke's functional equation and the average order of arithmetical functions. Acta Arith. 6 1960/1961 487-503. (Reviewer: E. Grosswald) 10.41
  • MR0111727 Chandrasekharan, K.; Mandelbrojt, S. On solutions of Riemann's functional equation. Bull. Amer. Math. Soc. 65 1959 358-362. (Reviewer: F. V. Atkinson) 10.00
  • MR0089222 Chandrasekharan, K.; Mandelbrojt, S. On Riemann's functional equation. Ann. of Math. (2) 66 (1957), 285-296. (Reviewer: S. Bochner) 10.1X
  • MR0086923 Chandrasekharan, K. On some problems in Fourier analysis. Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. III, pp. 85-91. Erven P. Noordhoff N.V., Groningen; North-Holland Publishing Co., Amsterdam, 1956. (Reviewer: Franti#ek Wolf) 42.2X
  • MR0085996 Chandrasekharan, K. Obituary: T. Vijayaraghavan. Math. Student 24 (1956), 251-267 (1957). 01.0X
  • MR0079034 Bochner, S.; Chandrasekharan, K. On Riemann's functional equation. Ann. of Math. (2) 63 (1956), 336-360. (Reviewer: H. D. Kloosterman) 10.1X
  • MR0069912 Chandrasekharan, K.; Minakshisundaram, S. A note on typical means. J. Indian Math. Soc. (N.S.) 18, (1954). 107-114. (Reviewer: S. Mandelbrojt) 40.0X
  • MR0055458 Chandrasekharan, K.; Minakshisundaram, S. Typical means. Oxford University Press, 1952. x+139 pp. (Reviewer: S. Mandelbrojt) 40.0X
  • MR1565284 DML Chandrasekharan, K.; Book Review: Lecons sur le calcul des coefficients d'une series trigonometrique. Bull. Amer. Math. Soc. 57 (1951), no. 2, 153-154.
  • MR0045234 Chandrasekharan, K. Fourier series, lattice points and Watson transforms. Math. Student 19, (1951). 1-11. (Reviewer: A. Zygmund) 42.4X
  • MR0043034 Chandrasekharan, K. Obituary: S. S. Pillai. J. Indian Math. Soc. (N.S.) Part A. 15, (1951). 1-10 (1 plate). 01.0X
  • MR0041960 Chandrasekharan, K. On the summation of multiple Fourier series. IV. Ann. of Math. (2) 54, (1951). 198-213. (Reviewer: A. Zygmund) 42.4X
  • MR0041896 Boas, R. P., Jr.; Chandrasekharan, K. Addendum: Derivatives of infinite order. Proc. Amer. Math. Soc. 2, (1951). 422. (Reviewer: S. Mandelbrojt) 27.0X
  • MR0034886 Bochner, S.; Chandrasekharan, K. Lattice points and Fourier expansions. Acta Sci. Math. Szeged 12, (1950). Leopoldo Fejer et Frederico Riesz LXX annos natis dedicatus, Pars B, 1-15. (Reviewer: A. Zygmund) 42.4X
  • MR0034809 Bochner, S.; Chandrasekharan, K. Summations over lattice points in #-space. (A supplementary note.) Quart. J. Math., Oxford Ser. (2) 1, (1950). 80. (Reviewer: N. G. de Bruijn) 10.0X
  • MR0032807 Bochner, S.; Chandrasekharan, K. Fourier series of L2-functions. Duke Math. J. 16, (1949). 579-583. (Reviewer: A. Zygmund) 42.4X
  • MR0031582 Bochner, S.; Chandrasekharan, K. Fourier Transforms. Annals of Mathematics Studies, no. 19. Princeton University Press, Princeton, N. J.; Oxford University Press, London, 1949. ix+219 pp. (Reviewer: H. Pollard) 42.4X
  • MR0027881 Bochner, S.; Chandrasekharan, K. Gauss summability of trigonometric integrals. Amer. J. Math. 71, (1949). 50-59. (Reviewer: A. Zygmund) 42.4X
  • MR0028338 Bochner, S.; Chandrasekharan, K. Summations over lattice points in #-space. Quart. J. Math., Oxford Ser. 19, (1948). 238-248. (Reviewer: N. G. de Bruijn) 10.0X
  • MR0027880 Chandrasekharan, K.; Szasz, Otto On Bessel summation. Amer. J. Math. 70, (1948). 709-729. (Reviewer: L. S. Bosanquet) 42.4X
  • MR0027309 Boas, R. P., Jr.; Chandrasekharan, K. Correction: Derivatives of infinite order. Bull. Amer. Math. Soc. 54, (1948). 1191. 27.0X
  • MR0027090 Chandrasekharan, K. On Fourier series in several variables. Ann. of Math. (2) 49, (1948). 991-1007. (Reviewer: A. Zygmund) 42.4X
  • MR0027089 Bochner, S.; Chandrasekharan, K. On the localization property for multiple Fourier series. Ann. of Math. (2) 49, (1948). 966-978. (Reviewer: A. Zygmund) 42.4X
  • MR0026145 Chandrasekharan, K. On the summation of multiple Fourier series. II. Proc. London Math. Soc. (2) 50, (1948). 223-229. (Reviewer: A. Zygmund) 42.4X
  • MR0026144 Chandrasekharan, K. On the summation of multiple Fourier series. I. Proc. London Math. Soc. (2) 50, (1948). 210-222. (Reviewer: A. Zygmund) 42.4X
  • MR0025527 Boas, R. P., Jr.; Chandrasekharan, K. Derivatives of infinite order. Bull. Amer. Math. Soc. 54, (1948). 523-526. (Reviewer: S. Mandelbrojt) 27.0X
  • MR0022925 Chandrasekharan, K.; Minakshisundaram, S. Some results on double Fourier series. Duke Math. J. 14, (1947). 731-753. (Reviewer: S. Bochner) 42.4X
  • MR0018249 Chandrasekharan, K. On multiple Fourier series. Proc. Indian Acad. Sci., Sect. A. 24, (1946). 229-232. (Reviewer: S. Bochner) 42.4X
  • MR0016157 Chandrasekharan, K. On the summation of multiple Fourier series. III. Bull. Amer. Math. Soc. 52, (1946). 474-477. (Reviewer: O. Szasz) 42.4X
  • MR0015347 Chandrasekharan, K. A further note on intuitionistic set-theory. Math. Student 13, (1945). 49-51. (Reviewer: O. Frink) 02.0X
  • MR0013444 Chandrasekharan, K. On Sturm-Liouville series. J. Indian Math. Soc. (N.S.) 8, (1944). 109-114. (Reviewer: W. W. Rogosinski) 42.4X
  • MR0011285 Chandrasekharan, K. Partially ordered sets and symbolic logic. Math. Student 12, (1944). 14-24. (Reviewer: G. Birkhoff) 09.1X
  • MR0011722 Chandrasekharan, K. On the canonical expression for a meromorphic function of finite order. J. Madras Univ. Sect. B. 15, (1943). 11-17. (Reviewer: R. P. Boas Jr.) 30.0X
  • MR0009064 Chandrasekharan, K. The absolute summability of series of eigenfunctions. J. Indian Math. Soc. (N.S.) 7, (1943). 25-30. (Reviewer: F. W. Perkins) 42.4X
  • MR0008848 Chandrasekharan, K. Bessel-summability of the product of two series. J. Indian Math. Soc. (N.S.) 7, (1943). 31-35. (Reviewer: F. W. Perkins) 40.0X
  • MR0009765 Chandrasekharan, K. Intuitionistic theory of linear order. Math. Student 10, (1942). 149-162. (Reviewer: O. Frink) 02.0X
  • MR0008847 Chandrasekharan, K. The absolute Bessel-summability of series. Bull. Calcutta Math. Soc. 34, (1942). 187-196. (Reviewer: R. P. Boas Jr.) 40.0X
  • MR0008845 Chandrasekharan, K. The second theorem of consistency for absolutely summable series. J. Indian Math. Soc. (N.S.) 6, (1942). 168-180. (Reviewer: R. P. Boas Jr.) 40.0X
  • MR0007373 Chandrasekharan, K. The logic of intuitionistic mathematics. Math. Student 9, (1941). 143-154. (Reviewer: O. Frink) 02.0X
  • MR0005757 Chandrasekharan, K. On Hadamard's factorization theorem. J. Indian Math. Soc. (N.S.) 5, (1941). 128-132. (Reviewer: R. P. Boas Jr.) 30.0X
Below is a picture of the "heilige Hallen" (sacred halls) as we had called this part of the ETH. In the H floor is the library, in the F floor the "Epsilon Schlauch" (a term used by students in mathematics as it reminds of a neighborhood of a function in C([0,1] in Arzela-Ascoli). The "epsilon-tube" contained coffee and snack machines. I essentially lived from those machines). My office G36.1 was in the G floor. There was also a large auditorium with a piano which was open for practice. The lecture halls are still the left or right on each floor. Chandra's number theory lecture took place in one of those large lecture halls. There were more than 100 students I guess in that class. About the halls: later, as graduate students, we have got access to the building which was otherwise closed. I would sometimes work through the winter break there (once writing with astronomers an article in a week). Especially around New Year, the sacred halls were empty and we (some very few graduate students) were for a few weeks the rulers of that large castle. The professors (priests), staff and custodians all gone, we, the monks of that monastery would be the gatekeepers.

Picture source: from a nice youtube clip of Jake Write exploring ETH. Oliver Knill, October 13, 2017,