The hypergeometric series of bradley and ramanujan is evaluated by means of the binomial convolutions of hagen and rothe, which presents, alternatively, a short proof of the recent result of bradley about ramanujan s. I beg to introduce myself few words on the genius ramanujan. In mathematics, ramanujan s master theorem named after mathematician srinivasa ramanujan is a technique that provides an analytic expression for the mellin transform of a function. In mathematics, ramanujans congruences are some remarkable congruences for the partition function p n. We explain how the subject started, the known explicit constructions of such graphs, the analogy between the spectral analysis of graphs and riemannian manifolds, the distribution of the spectra. Keep in mind that a squarefree rational integer is an element a2z such that the prime factorization of a has no squared factors.
Srinivasa ramanujan was a largely selftaught pure mathematician. In mathematics, a ramanujan prime is a prime number that satisfies a result proven by srinivasa ramanujan relating to the primecounting function. In mathematics, the hardy ramanujan theorem, proved by g. In his famous letters of 16 january 19 and 29 february 19 to g.
Probably ramanujan s life has no parallel in the history of human thought. Introduction srinivasa ramanujan, hailed as one of the greatest mathematicians of this century, left behind an incredibly vast and formidable amount of original work, which. Life and work of the mathemagician srinivasa ramanujan k. Pdf a note on mathematics genius srinivasa ramanujan, frs.
Hardy is his formula for the number of partitions of a positive integer n, the famous hardy ramanujan asymptotic formula for the partition problem. Journal of number theory 25, 119 1987 a formula of s. But the reason these graphs are important for cryptography is that nding paths in these graphs, i. Ramanujans series for 1 a survey department of mathematics. However, they are all in the spirit of his mathematics. In this paper we compute the highdimensional laplace spectrum of ramanujan triangle complexes. Where can i find a list of the unsolved problems of ramanujan. Ramanujan introduced a technique, known as ramanujan s master theorem, which provides an explicit expression for the mellin transform of a function in terms of the analytic continuation of its taylor coefficients. The author was partially supported by iami cnr, milano, 1994.
The man who knew infinity ramanujan college of management. Ramanujan school of mathematics and computer science department of computer science master of science curriculum effective from the academic year 20092010 note. He recorded his results without proof which has been edited later by bruce c. Generating functions, partitions, and qseries modular forms applications generating functions, partitions, and qseries. As murtys survey paper notes, ramanujan graphs fuse diverse branches of pure mathematics, namely, number theory, representation theory, and algebraic. Pdf when we pause to reflect on ramanujans life, we see that there were certain events that seemingly were. Ramanujan s pages of theorems were like an alien forest whose trees were familiar enough to call trees.
Assume function f x fx f x has an expansion of the form. Ramanujan notebooks iii riemann hypothesis solutions. Later some theorems were established in theory of elliptic curves which involves this fascinating number. In mathematics, ramanujans master theorem named after srinivasa ramanujan is a technique that provides an analytic expression for the mellin transform of an analytic function. Transformations of ramanujans summation formula and its applications chandrashekar adiga1 and n. Sitaramachandrarao department of mathematics, university of toledo, toledo, ohio 43606 communicated by n.
Some of ramanujans most famous theorems are found in chapter 16. Ramanujans most fascinating theorems in the lost notebook is the. Srinivasa ramanujan 18871920 ramanujan was one of the greatest mathematical geniuses ever to emerge from india. Ramanujan s formula needs to be slightly corrected, but what is remarkable is that such a formula exists. Ramanujan in this paper, we discuss various equivalent formulations for the sum of an infinite series considered by s. Ramanujan published a paper 62 entitled, squaring the circle in the jour. He studies their structure, distribution and special forms. Ramanujan hailed as an alltime great mathematician, like euler, gauss or jacobi, for his natural genius, has left behind 4000 original theorems, despite his lack of formal education and a short lifespan. Though he had almost no formal training in pure mathematics, he made substantial contributions to. Taekyun kim 3 institute of science education, kongju national university, kongju 314701, korea. The partition function and ramanujan congruences oregon state. Each chapter of journey through genius has three primary components. Pdf contributions of srinivasa ramanujan to number theory. An introduction to classical modular forms edray herber goins ma 5980061427988.
Ramanujan graphs have extremal spectral properties, which imply a remarkable combinatorial behavior. Srinivasa rao the institute of mathematical sciences, chennai 600 1. Formulas of ramanujan for the power series coefficients of certain. Roughly speaking, this means that most numbers have about this number of distinct prime factors. By combining this and the previous result and solving for w. In the irish times column this week tm010, we tell how a collection of papers of srinivasa ramanujan turned up in the wren library in cambridge and set the mathematical world ablaze. Contents 1 introduction 1 2 later history 3 3 fundamental lemma 3 4 ivory. The mathematician srinivasa ramanujan discovered the congruences. Also it welcomes open problems in the line of the aim of this udt for possible publication in this section. Hindered by poverty and illhealth, his highly original work has considerably enriched number theory. Shaun cooper massey university, auckland askey 80 conference. Magic squares, theory of partitions, ramanujans contribution to the concept of highly composite numbers, expressions for. Ramanujan did not leave us proofs of the thousands of theorems he recorded in his notebooks, and so it cannot be claimed that many of the proofs given in this book are those found by ramanujan.
The purpose of this paper is to introduce some of the contributions of srinivasa ramanujan to number theory. Trench 1 foreword thisisarevisedandextendedversionofsection6. Code course title hs credits 1 411 design of algorithms h 3 2 4 computer architecture and organization h 3. These are special cases of ramanujan s theorem appearing as entry 4 on page 108, chapter 12 of his second notebook.
Srinivasa ramanujan was born on december 22, 1887 in the. Ramanujans theories of elliptic functions to alternative. This phrase actually means that ramanujan was such a natural genius that his contributions to mathematics could reach infinity if he would have lived forever. The formula has been used in statistical physics and is als. Some of those formulas were analyzed by hardy 3, 5, pp. Ramanujans formula for the riemann zeta function extended. Origins and definition edit in 1919, ramanujan published a new proof of bertrands postulate which, as he notes, was first proved by chebyshev. Combining the two formulas to eliminate the term k. All course codes are to be preceded with coms first semester s. Classical modular forms department of mathematics purdue university. Ramanujan s formula for the riemannzeta function is one of his most celebrated. In spectral graph theory, a ramanujan graph, is a regular graph whose spectral gap is almost as large as possible see extremal graph theory. Theorem of the day the hardy ramanujan asymptotic partition formula for n a positive integer, let pn denote the number of unordered partitions of n, that is.
Ramanujan primes and bertrands postulate jonathan sondow 1. Ramanujan studied the highly composite numbers also which are recognized as the opposite of prime numbers. Srinivasa ramanujan s contributions in mathematics dharminder singh1, arun kumar chopra2, sukhdeep singh bal3 1,2asst. What are the real world applications of ramanujans findings. The great theorems on the pages ahead span more than 2300 years of human history.
Ramanujan graphs constructed in the literature, which we shall refer to below, have been all simple graphs. Page from ramanujans notebook stating his master theorem. Ramanujan s notebooks ramanujan s nested radicals mock theta function and ramanujan theta function landau ramanujan constant ramanujan conjecture ramanujan prime ramanujan soldner constant ramanujan s sum roger ramanujan identities master theorem some properties of bernoullis numbers. Ramanujan s mock theta functions and their applications dapr es zwegers and bringmannono by don zagier introduction one of the most romantic stories in the history of mathematics is that of the friendship between hardy and ramanujan. Leaving aside the questions of convergence of these in. Anitha 2 department of studies in mathematics university of mysore, manasagangotri mysore570006, india.
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