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Research Publications of Ramanujan

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Research Publications of Ramanujan Appendix A Research Publications of Ramanujan Papers Written by S. Ramanujan 1. Some Properties of Bernoulli’s numbers Journal of the Indian Mathematical Society,3(1911) 219–234. 2. On Question 330 of Prof. Sanjana Journal of the Indian Mathematical Society,4(1912) 59–61. 3. Note on a set of simultaneous equations Journal of the Indian Mathematical Society,4(1912) 94–96. 4. Irregular numbers Journal of the Indian Mathematical Society,5(1913) 105–106. 5. Squaring the circle Journal of the Indian Mathematical Society,5(1913) 132. 6. Modular equations and approximations to π Quarterly Journal of Mathematics,45(1914) 350–372. x tan−1t 7. On the integral 0 t dt Journal of the Indian Mathematical Society,7(1915) 93–96. 8. On the number of divisors of a number Journal of the Indian Mathematical Society,7(1915) 131–133. 9. On the sum of the square roots of the first n natural numbers Journal of the Indian Mathematical Society,7(1915) 173–175. 3 n=∞ + x 10. On the product n=0 1 a+nd Journal of the Indian Mathematical Society,7(1915) 209–211. 11. Some definite integrals Messenger of Mathematics,44(1915) 10–18. 12. Some definite integrals connected with Gauss’s sums Messenger of Mathematics,44(1915) 75–85. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 231 K. Srinivasa Rao, Srinivasa Ramanujan, https://doi.org/10.1007/978-981-16-0447-8 232 A Research Publications of Ramanujan 13. Summation of certain series Messenger of Mathematics,44(1915) 157–160. 14. New expressions for Riemann’s functions ξ(s) and (t) Quarterly Journal of Mathematics,46(1915) 253–260. 15. Highly composite numbers Proceedings of the London Mathematical Society,2, 14 (1915) 347–409. 16. On certain infinite series Messenger of Mathematics,45(1916) 11–15. 17. Some formulae in the analytic theory of numbers Messenger of Mathematics,45(1916) 81–84. 18. On certain arithmetical functions Transactions of the Cambridge Philosophical Society,22, No.9 (1916) 159– 184. 19. A series for Euler’s constant γ Messenger of Mathematics,46(1917) 73–80. 20. On the expression of a number in the form ax2 + by2 + cz2 + du2 Proceedings of the Cambridge Philosophical Society,19(1917) 11–21. 21. On certain trigonometrical sums and their applications in the theory of numbers Transactions of the Cambridge Philosophical Society,22, No. 13 (1918) 259–276. 22. Some definite integrals Proceedings of the London Mathematical Society,2, 17(1918) Records for 17 January 1918. 23. Some definite integrals Journal of the Indian Mathematical Society,11, (1919) 81–87. 24. A proof of Bertrand’s postulate Journal of the Indian Mathematical Society,11 (1919) 181–182. 25. Some properties of p(n), the number of partitions of n Proceedings of the Cambridge Philosophical Society,19(1919) 207–210. 26. Proof of certain identities in combinatory analysis Proceedings of the Cambridge Philosophical Society,19, (1919) 214–216. 27. A class of definite integrals Quarterly Journal of Mathematics,48(1920) 294–310. 28. Congruence properties of partitions Proceedings of the London Mathematical Society,2, 18 (1920) Records for 13 March 1919. 29. Algebraic relations between certain infinite products Proceedings of the London Mathematical Society,2, 18 (1920) Records for 13 March 1919. 30. Congruence properties of partitions Mathematische Zeitschrift,9(1921) 147–153. A Research Publications of Ramanujan 233 Papers Written by S. Ramanujan with G.H. Hardy 1. Une formulae asymptotique pour le nombre des partitions de n Comptes Rendus, 2 January 1917. 2. Proof that almost all numbers n are composed of about log log n factors Proceedings of the London Mathematical Society,2, 16 (1917) Records for 14 December 1916. 3. Asymptotic formulae in combinatory analysis Proceedings of the London Mathematical Society,2, 16 (1917) Records for 1 March 1917. 4. Asymptotic formulae for the distribution of integers of various types Proceedings of the London Mathematical Society,2, 18 (1920) 112–132. 5. The normal number of prime factors of a number n Quarterly Journal of Mathematics,48 (1917) 76–92. 6. Asymptotic formulae in combinatory analysis Proceedings of the London Mathematical Society,2, 17 (1918) 75–115. 7. On the coefficients in the expansions of certain modular functions Proceedings of the Royal Society,A,95 (1918) 144–155. These 37 research publications of Ramanujan are reprinted in the Collected Papers of Srinivasa Ramanujan, edited by G.H. Hardy, P.V. Seshu Aiyar and B.M. Wilson and published first by the Cambridge University Press (1927) and later by Chelsea (1962). The questions posed and the answers provided by Ramanujan, 59 in all, in the Journal of the Indian Mathematical Society follow the research papers in the Collected Papers. Besides the Preface, the editors include two biographical Notices, one by P.V. Seshu Aiyar and R. Ramachandra Rao and another by G.H. Hardy, at the beginning of the Collected Papers. Notes on the research publications and some extracts from Ramanujan’s letters to G.H. Hardy are in two Appendices, at the end of the Collected Papers. Appendix B Wren Library Card Catalogue and Papers of Ramanujan With the help of Professor Robert A. Rankin, the Wren Library has catalogued the papers of Ramanujan. These have the reference addresses as Add.Ms.a.94, with a superscript which refers to the specific document, and as Add.Ms.b.100–107C. The papers that are available with reference Add.Ms.a.941−18 are contained in a large sized (A3-size) 2 inches high cardboard box. The reference Add.Ms.b. 100– 107C refers to the five bound volumes (foolscap size), and, in addition to these, the papers and letters are preserved in two large (A3) sized thick cardboard boxes. In this Appendix are given the details of the Card Catalogue on Ramanujan, as well as the details of the Ramanujan MSS as maintained in the Archives of the Wren Library of Trinity College, Cambridge University. In fact, a part of the Appendix Card Catalogue is what is contained in Supplement II of Venkatachaliengar’s article, as “A list of all the Ramanujan - material at the Trinity College Cambridge given to Professor C.T. Rajagopal by Rankin”. The Card Catalogue in the Wren Library of Trinity College: 1. AIYAR A(P Venkatesvara Seshu) see AYYAR (P Venkatesvara Seshu) 2. ANDERSON (Sir Hugh Kerr) see HARDY (G.H.) 3. AYYAR (P Venkatesavara Seshu) Letter to G.H. Hardy, 23 March 1928 Add.Ms.a.941(31) 4. AYYAR (P Venkatesavara Seshu) Letter to B.M. Wilson, 8 July 1925 Add.Ms.a.9412(49) 5. AYYAR (P Venkatesavara Seshu) Letter to B.M. Wilson, 22 April 1926 Add.Ms.a.9412(58) © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 235 K. Srinivasa Rao, Srinivasa Ramanujan, https://doi.org/10.1007/978-981-16-0447-8 236 B Wren Library Card Catalogue and Papers of Ramanujan 6. BERWICK (William Edward Hodgson) Letter [to G.N. Watson ?], 24 Oct. 1916 Add.MS.a.945 7. CAMBRIDGE UNIVERSITY PRESS see HARDY (G.H.) RAMANUJAN (S.) 8. CHANDRASEKHAR (Subrahmanyan) Letter to G.H. Hardy, 4 Aug. 1937 [and 28 Dec. 1937] Add.MS.a.948 9. CLARK (Edward Mellish) Letters to G.H. Hardy, 7–25 May 1920 Add.MS.a.94(12−24) 10. DEWSBURY (Francis) Letters to G.H. Hardy, 5 March 1918–15 Jan. 1929 Add.MS.a.94(13−17) 11. DEWSBURY (Francis) Letter to G.H. Hardy, 13 July 1922 Add.MS.a.9412(40) 12. DEWSBURY (Francis) Letter to G.H. Hardy, 2 Aug. 1923 Add.MS.a.9412(25) 13. DEWSBURY (Francis) Letter to G.H. Hardy, 6 Aug.–13 Sept. 1923 Add.MS.a.94(27–29) 14. FOWLER (Sir Ralph Howard) Letter to G.H. Hardy, 9 June [1922] Add.Ms.a.9412(34) 15. HARDY (Godfrey Harold) “G.H. Hardy in account with the Syndics of the Cambridge University press: Collected Papers of S. Ramanujan. Statement of Account from 1 January 1928 to 31 December 1928”, April 1929. Add.Ms.a.9412(77) 16. HARDY (Godfrey Harold) Letter to S. Ramanujan, 6 Feb. 1918, on summing “partition series” for r θ3 (τ)(r odd) Add.MS.a.9416 17. HARDY (Godfrey Harold) Letter to G.N. Watson, [April 1929 ] Add.MS.a.9412(76) 18. HARDY (Godfrey Harold) Letter to G.N. Watson, [n.d.1] Add.MS.a.943 1The abbreviation n.d. stands for not dated. B Wren Library Card Catalogue and Papers of Ramanujan 237 19. HARDY (Godfrey Harold) Letter to B.M. Wilson, [5 June 1925] Add.Ms.a.9412(32) 20. HARDY (Godfrey Harold) Letter to B.M. Wilson, [10 June 1925] Add.MS.a.9412(41−43) 21. HARDY (Godfrey Harold) Envelope addressed to B.M. Wilson, 14 Oct. 1925 Add.MS.a.9412(56) 22. HARDY (Godfrey Harold) Letter to B.M. Wilson, 15 Oct. [1925] Add.MS.a.9412(52) 23. HARDY (Godfrey Harold) Letter to B.M. Wilson, [Jan./Feb. 1927] Add.MS.a.9412(62) 24. HARDY (Godfrey Harold) Letters to B.M. Wilson, [n.d.2] Add.MS.a.9412(5−8) 25. HARDY (Godfrey Harold) Letters to B.M. Wilson, [n.d.] Add.MS.a.9412(20−21) 26. RAMANUJAN (Srinivasa) Letter on mock theta functions. [This appears to be the letter of 12 January 1920, partly reproduced on pp. 354–355 of the Collected Papers.] 27. HARDY (Godfrey Harold) “Memorandum of agreement made this 12 August 1925 between Professor G.H.Hardy,NewCollege,Oxford...andtheSyndics of the University Press Cambridge ...” [concerning S.
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