Collected Papers of Srinivasa Ramanujan de Srinivasa Ramanujan

Collected Papers of Srinivasa Ramanujan

Srinivasa Ramanujan

G. H. Hardy

P. V. Seshu Aiyar

B. M. Wilson

Originally published in 1927, this book presents the collected papers of the renowned Indian mathematician Srinivasa Ramanujan (1887–1920), with editorial contributions from G. H. Hardy (1877–1947). Detailed notes are incorporated throughout and appendices are also included. This book will be of value to anyone with an interest in the works of Ramanujan and the history of mathematics.

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Cuprins

Preface; Notice P. V. Seshu and R. Bamachaundra Rao; Notice G. H. Hardy; Part I. Papers: 1. Some properties of Bernoulli's numbers; 2. On question 330 of Prof. Sanjana; 3. Note on a set of simultaneous equations; 4. Irregular numbers; 5. Squaring the circle; 6. Modular equations and approximations to π; 7. On the integral [...]; 8. On the number of divisors of a number; 9. On the sum of the square roots of the first n natural numbers; 10. On the product [...]; 11. Some definite integrals; 12. Some definite integrals connected with Gauss's sums; 13. Summation of a certain series; 14. New expression for Riemann's functions [...]; 15. Highly composite numbers; 16. On certain infinite series; 17. Some formulae in the analytic theory of numbers; 18. On certain arithmetical functions; 19. A series of Euler's constant y; 20. On the expression of a number in the form of ax2+by2+cz2+du2; 21. On certain trigonometrical sums and their applications in the theory of numbers; 22. Some definite integrals; 23. Some definite integrals; 24. A proof of Bertrand's postulate; 25. Some properties of p (n), the number of partitions of n; 26. Proof of certain identities in combinatory analysis; 27. A class of definite integrals; 28. Congruence properties of partitions; 29. Algebraic relations between certain infinite products; 30. Congruence properties of partitions; 29. Algebraic relations between certain infinite products; 30. Congruence properties of partitions; Part II. Papers Written in Collaboration with G. H. Hardy: 31. Une formule asymptotique pour le nombre des partitions de n; 32. Proof that almost all numbers n are composed of about log log n prime factors; 33. Asymptotic formulae in combinatory analysis; 34. Asymptotic formulae for the distribution of integers of various types; 35. The normal number of prime factors of a number n; 36. Asymptotic formulae in combinatory analysis; 37. On the coefficients in the expansions of certain modular functions; Questions and solutions; Appendix 1. Notes on the papers; Appendix 2. Further extracts from Ramanujan's letters to G. H. Hardy.

Recenzii

'[The book] is introduced by a pair of notes which are sources of wonderful information about Ramanujan in their own right, both as regards his life and his mathematics. After that it is all about his mathematics: thirty-seven articles on number theory, infinite series, integrals, and combinatorics. It is all stunning, both by virtue of the content of these articles and because of the idiosyncrasy of their author.' Michael Berg, MAA Reviews