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Collected papers of Srinivasa Ramanujan

1928; Oxford University Press; Volume: CLVI; Issue: feb11 Linguagem: Inglês

10.1093/nq/clvi.feb11.107

1471-6941

Srinivasa Ramanujan Aiyangar, G. H. Hardy, P. V. Seshu Aiyar, B. M. Wilson,

Mathematical functions and polynomials

Some properties of Bernoulli's numbers On Question 330 of Prof. Sanjana Note on a set of simultaneous equations Irregular numbers Squaring the circle Modular equations and approximations to $\pi$ On the integral $\int^x_0\frac{\tan^{-1}t} {t}dt$ On the number of divisors of a number On the sum of the square roots of the first $n$ natural numbers On the product $\prod^{n=\infty}_{n=0}[1+(\frac{x} {a+nd})^3]$ Some definite integrals Some definite integrals connected with Gauss's sums Summation of a certain series New expressions for Riemann's functions $\xi(s)$ and $\Xi(t)$ Highly composite numbers On certain infinite series Some formulae in the analytic theory of numbers On certain arithmetical functions A series for Euler's constant $\gamma$ On the expression of a number in the form $ax^2 + by^2 + cz^2 +du^2$ On certain trigonometrical sums and their applications in the theory of numbers Some definite integrals Some definite integrals A proof of Bertrand's postulate Some properties of $p(n)$, the number of partitions of $n$ Proof of certain identities in combinatory analysis A class of definite integrals Congruence properties of partitions Algebraic relations between certain infinite products Congruence properties of partitions Une formule asymptotique pour le nombre des partitions de $n$ Proof that almost all numbers $n$ are composed of about $\log \log n$ prime factors Asymptotic formulae in combinatory analysis Asymptotic formulae for the distribution of integers of various type The normal number of prime factors of a numbers $n$ Asymptotic formulae in combinatory analysis On the coefficients in the expansions of certain modular functions Questions and solutions Appendix I: Notes on the papers Appendix II: Further extracts from Ramanujan's letters to G. H. Hardy.