March 8, 20 ramanujan educational institutions palwal india 4 child prodigy learned collegelevel mathematics by age 11, and generated his own theorems in number theory and bernoulli numbers by age including independently rediscovering eulers identity. Ramanujan is recognized as one of the great number theorists of the twentieth century. A generalization of ramanujans congruence to modular forms of prime level. An easy proof of the rogersramanujan identities open archive. This book provides an introduction to these two important subjects and to some of the topics in number theory that are inextricably intertwined with them, including the theory of partitions, sums of squares and triangular numbers, and the ramanujan tau function.
Hirschhorn, some interesting qseries identities, ramanujan journal 3612. The page has a collection of about sixty problems which have appeared in the journal of the indian mathematical society. A valuable resource for mathematicians, the journal provides an international forum for. Hardy later told the nowfamous story that he once visited ramanujan at a nursing home, telling him that he came in a taxicab with number 1729, and saying that it seemed to him a rather dull numberto which ramanujan replied. Ramanujan had noted down the results of his researches, without proofs, as in a synopsis of elementary results, a book on pure mathematics, by g. Ramanujans mock theta functions and some recent developments. In press, journal preproof, available online 22 may 2020.
The journal of number theory jnt features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. Hirschhorn, ramanujans partition congruences, discrete mathematics 1 1994 3555. Ono served as associate producer and consultant for the forthcoming film on the life and work of ramanujan, the man who knew infinity. Hirschhorn, some interesting qseries identities, ramanujan journal 361 2.
Duration july 25 august 6, 2016 number of participants for the course will be limited to fifty. Pdf the purpose of this paper is to introduce some of the. Hardyramanujan number once hardy visited to putney where ramanujan was hospitalized. The rstirling numbers of the first kind in terms of the mobius function. Highly composite numbers by srinivasa ramanujan universite. While in school, he was gifted george schoobridge carrs synopsis. H hardy summed up his contibution in following words, the limitations of his knowledge were as startling as its profundity. The man who knew infinity ramanujan college of management. Rms lecture notes series ramanujan mathematical society. The objective of this paper is to prove two of these modular equations by employing the theory of theta functions identities stated by ramanujan. Hypergeometric and basic hypergeometric series qseries.
Journal of the ramanujan mathematical society is published by the the ramanujan mathematical society after ramanujan mathematical society was founded in 1985, the starting of its journal journal of the ramanujan mathematical society jrms followed as a sequitur in 1986. Rsmams ramanujan society of mathematics and mathematical. The cayley graphs associated with some quasiperfect lee codes are ramanujan graphs. Lutz journal of number theory 186 2018 115 3 table 2. The purpose of this paper is to introduce some of the contributions of srinivasa ramanujan to number theory. Most downloaded journal of number theory articles elsevier. At the end of the twopage published paper, ramanujan derived a generalized result, and that is. Reeder and yu have recently given a new construction of a class of supercuspidal representations called epipelagic representations m. The main aim of the society is the promotion of mathematics educations and its applications in different field of sciences and also to create interest among students, research scholars and young teachers towards mathematics. Journal of number theory a short and simple proof of. This indicator counts the number of citations received by documents from a journal and divides them by the total number of documents published in that journal. Some of those formulas were analyzed by hardy 3, 5, pp.
The chart shows the evolution of the average number of times documents published in a journal in the past two, three and four years have been cited in. International journal of number theory online ready. In mathematics, a ramanujan prime is a prime number that satisfies a result proven by srinivasa ramanujan relating to the primecounting function origins and definition. A survey of ramanujan expansions international journal. Contents 5 the set of all distribution functions of a sequence xn mod 1 will be denoted by gxn mod 1. Hirschhorn journal of number theory 00 20 14 4 11 m. The last expressions for the socalled rogersramanujan continued fraction appeared in ramanujans.
Variations of lehmers conjecture for ramanujans taufunction. Srinivasa ramanujan 18871920 was an indian mathematician who made great and original contributions to many mathematical fields, including complex analysis, number theory, infinite series, and continued fractions. This journal publishes papers of the highest quality in all areas of mathematics influenced by ramanujan, including. While i was browsing through them at random, i came across this, which i recognized as the brocardramanujan problem. As one who mooted the idea of starting the journal, the mantle of editorinchief fell naturally upon professor k. We shall identify the notion of the distribution of a sequence xn mod 1 with the set gxn mod 1, i. Highly composite numbers by srinivasa ramanujan springerlink. A formula of ramanujan in the theory of primes hardy. During his school days, he impressed his classmates, senior students and teachers with his extraordinary intuition and astounding pro. Volume 42 of hrj is dedicated to the fond memory of alan baker, who would have turned 80 in august, 2019. One feels that ramanujan is ready to leave the subject of highly composite numbers, and to come back to another favourite topic, identities. With the formation of the ramanujan mathematical society rms in 1985, the starting of the journal of the ramanujan mathematical society jrms, which was in the offing, was realised in 1986.
Sharath journal of number theory 3 20 437445 by using the theory of modular forms. As the reader might have taken from the name, ant expands and. Ramanujan theta function ramanujans sum rogersramanujan identities. Additionally, he serves as editorinchief for several journals, including research in the mathematical sciences and research in number theory, and he is an editor of the ramanujan journal. Centre for advanced study in mathematics, panjab university, chandigarh160014, india. Unlimited viewing of the articlechapter pdf and any. We explicitly calculate the local langlands correspondence for certain families of epipelagic representations of unitary groups.
Ramanujan, robin, highly composite numbers, and the riemann. Life and work of the mathemagician srinivasa ramanujan. Ramasamy, ramanujan s equation, journal of ramanujan mathematical society. You should attend if you are a student or faculty from academic institution interested to learn the influence of ramanujan in number theory.
Definitions of hypergeometric series and elliptic integrals. India celebrated the mathematics year in 2012 and ramanujan society of mathematics and mathematical sciences rsmams was established in the same year. Hirschhorn, a birthday present for ramanujan, american mathematical monthly 97 1990 398400. The work that ramanujan did in his brief professional life a century ago has spawned whole new areas of mathematical investigation, kept top mathematicians busy for their whole professional lives, and is finding applications in computer science, string theory, and the mathematical basis of black hole physics. Hardyramanujan journal entered the episciences platform in 2014 and will publish its future volumes through this free online open access portal. The remarkable discoveries made by srinivasa ramanujan have made a great impact on several branches of mathematics, revealing deep and fundamental connections. Yu, epipelagic representations and invariant theory, j. Ramanujan conjecture ramanujan prime ramanujan soldner constant. Here now is the first book to provide an introduction to his work in number theory. Littlewood, two worldclass mathematicians at cambridge, and enjoyed an extremely fruitful period of collaboration with them.
These formulae are among some problems posed by ramanujan in the journal of the indian mathematical society. The set gxn mod 1 has the following fundamental properties for every sequence xn mod 1. If a complexvalued function has an expansion of the form. Here was a man who could work out modular equations and theoremsto orders unheard o. Hardyramanujan number srinivasa ramanujan, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. It unfolds in a story of interesting connections as is described below. Each person applying for membership must be proposed by one member and seconded by another member both of whom are expected to have personal knowledge of the applicant and hisher interests in the object of the society. This paper summarizes the development of ramanujan expansions of arithmetic functions since ramanujans paper in 1918, following carmichaels meanvaluebased concept from 1932 up to 1994. Mystery of ramanujan number online international journal. Journal of number theory vol 159, pages 1436 february 2016. Our july insights column was inspired by the mathematics of the phenomenal 20thcentury number theorist srinivasa ramanujan, whose romantic and tragic life story was the subject of the recent film the man who knew infinity. The remarkable discoveries made by srinivasa ramanujan have made a great impact on.
Ramasamy, ramanujans equation, journal of ramanujan mathematical society. Then, without assuming the prime number theorem, we can show that the maximum order of q2n is. A new technique, based on the concept of related arithmetic functions, is introduced that leads to considerable extensions of preceding results on ramanujan. What are the mathematical contributions of ramanujan. The most downloaded articles from journal of number theory in the last 90 days.
Hardy arranged for ramanujan to come to england, and the rest is history. Read the latest articles of journal of number theory at, elseviers leading. Ramanujan was sent to kangeyam primary school in kumbakonam at the age of seven. Padmanabhan, then director of the ramanujan institute for advanced study in mathematics in chennai, took charge as the. Elementary number theory, group theory and ramanujan. The hardyramanujan journal the ramanujan journal uniform distribution theory algebra and number theory. Journal of number theory vol 193, pages 1424 december.
Here p denotes an odd prime number and g a primitive root modulo p. But it was not the whole work on the subject, and in the lost notebook and other unpublished papers, one can find a manuscript, handwritten by ramanujan, which is the continuation of the paper published by the london mathematical society. Ramanujans notebooks the history of the notebooks, in brief, is the following. Highly composite numbers 121 75, the number of representations of n by some other quadratic forms is considered, but no longer its maximal order. In 1915, the london mathematical society published in its proceedings a paper of ramanujan entitled highly composite numbers. Three puzzles inspired by ramanujan quanta magazine. International journal of number theory world scientific. In 1919, ramanujan published a new proof of bertrands postulate which, as he notes, was first proved by chebyshev. As was usual with ramanujan, he had a general formula hidden.
In his famous letters of 16 january 19 and 29 february 19 to g. My search for ramanujan how i learned to count ken ono. Contributions of srinivasa ramanujan to number theory. Ram murty journal of number theory 3 we recall the vital remark we made in 8, in order to get, for fairly general f and g. In the wikipedia page on ramanujan, there is a link to a collection of problems posed by him. Most of ramanujans work in number theory arose out of \q\series and theta functions.
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