\documentclass[11pt]{article} \usepackage{placeins} \usepackage{amsbsy,amsmath,amsfonts,amsthm} \usepackage{epsfig} \usepackage{graphicx} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{example}[theorem]{Example} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}{Remark} \title{The Man Who Knew Infinity} \author{[Enter your Name] \\MATH-199 Math/Stat Seminar\\Assignment \# 5(The second latex assignment)} \date{$17^{th}$ October, 2019} \begin{document} \maketitle Srinivasa Ramanujan (1887-1920) was an Indian mathematician that had almost no formal training in mathematics,but made notable contributions to the fields of continued fraction, infinite series,number theory, and mathematical analysis. Initially working in isolation in India, he quickly became recognized by fellow Indian mathematicians. After becoming known to the wider mathematical community, Ramanujan was mentored by English mathematician G.H.Hardy (1877-1947), a professor at Cambridge ,first through correspondence and second through visits to England to converse with Hardy and other top mathematicians of the time. Throughout his lifetime, Ramanujan produced nearly 4,000 results (some of which were previously known). Nearly all of the results have been verified to be correct. Today, most mathematicians consider Ramanujan one of the greatest Indian mathematicians in history. \section{Early Life} Srinivasa Ramanujan (see Figure\ref{1}) was born on 22 December 1887 in Erode, located in the southern Indian state of Tamil Nadu.\par He began to focus on mathematics at an early age, and, at the age of about fifteen, borrowed a copy of G.s.Carr's \textit{synopsis of pure and applied mathematics}\cite{george01},which served as his primary source of learning mathematics. At about the time Ramanujan entered college, he began to record his mathematical discoveries in notebooks.\par Living in poverty with no means of financial support, suffering at times from serious illnesses, and working in isolation, Ramanujan devoted his efforts to mathematics and continued to record his discoveries without proofs in notebooks for the next six years. \FloatBarrier \begin{figure} \centering \includegraphics[width=50mm]{A5_Ramanujan} \caption{Photo of Ramanujan taken in his late twenties} \label{1} \end{figure} \section{Elementary Mathematics} Many of Ramanujan's discoveries can be appreciated by those with only a knowledge of high school algebra. Chapter 2 in the second notebook, the unorganized portions of the second and third notebooks, and the problems that Ramanujan submitted to the \textit{Journal of the Indian mathematics society }\cite{srinvasa} are excellent sources for these gems.\par Ramanujan enjoyed finding equal sums of powers. For example, in \cite{Bruce1},he shows if $a+b+c = 0$,then \begin{eqnarray} 2(ab+ac+bc)^{4} =a^{4}(b-c)^{4} +b^{4}(a-c)^{4}+c^{4}(a-b)^{4}. \end{eqnarray} In fact, in his third notebook [5, p.96], Ramanujan recorded similar formulas for $2(ab+ac+bc)^{2n}$, when $n\in\left\lbrace 1,2,3,4\right\rbrace$ , and wrote "and so on" to indicate that he possessed a general procedure for finding such formulas.\par Ramanujan was fond of stating intriguing formulas such as \begin{eqnarray} 2 \text{sin}(\frac{\pi}{18}) = \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2-...}}}} \end{eqnarray} or \begin{eqnarray} \sqrt[3]{\text{cos}40^{0}}+\sqrt[3]{\text{cos}80^{0}}-\sqrt[3]{\text{cos}20^{0}} = \sqrt[3]{\frac{3}{2}\left(\sqrt[3]{9}-2\right)}. \end{eqnarray} In most instances, these are special cases of more general theorems that he established.\newpage \section{His Notebooks} To provide a feeling of Ramanujan's notebooks, we produce here(see Figure \ref{11}) a page from his third notebook, chapter XVII\cite{Bruce}. There is no particular reason for choosing this page except that it contains an entry which has become a 'folklore' which illustrates the attachment Ramanujan had to numbers prompting British mathematician John E, Littlewood (1885-1977) to state that to Ramanujan every number is a personal friend.\par There are two parts shown in this provided page. The first part is concerning the geometrical construction of a square whose are is equal to that of a given circle. Here he gives a geometrical construction for finding the length of the side of a square whose area equals that of the circle. He also reproduced this and another geometrical construction for $\pi$ in his later paper on "Modular equations and approximations to $\pi$ ". In this paper he deduced several approximations and formulae for $\pi$ such as \begin{eqnarray} \pi \approx \frac{63}{25}\left( \frac{17+15\sqrt{5}}{7+15\sqrt{5}} \right), \label{15} \end{eqnarray} \begin{eqnarray} \pi \approx 12\sqrt{190}\text{log}\left\lbrace (2\sqrt{2}+\sqrt{10})(3+\sqrt{10}) \right\rbrace , \label{19} \end{eqnarray} and \begin{eqnarray} \frac{1}{\pi} = \frac{2\sqrt{2}}{9801}\sum_{n=0}^{\infty} \frac{(4n)!(1103+26390n)}{(n!)^{4}396^{4n}}. \label{20} \end{eqnarray} The expression in (\ref{15}) gives the value of $\pi$ accurate to 9 decimal places and the one in (\ref{19}) has an accuracy of 18 decimal places. Ramanujan asserted that (\ref{20}) is the most intriguing one because it is a very "rapidly convergent" series.In 1986, two computer scientists used aversion of Ramanujan's formula to calculate $\pi$ to 17 million digits and found that the formula converges to the exact value with far greater efficiency than any previous method. This success proved that Ramanujan's insight was correct. \section{The Hardy-Ramanujan Number} one of the most famous stories related to Ramanujan is the \textit{Hardy-Ramanujan number}. Hardy was visiting Ramanujan in a hospital, and, as told by Hardy himself: \begin{figure} \centering \includegraphics[width=60mm]{A5_notebookpage} \caption{ Ramanujan's handwritten page from his \textit{Notebook}.} \label{11} \end{figure} \FloatBarrier "I remember once going to see him when he was ill at Putney.I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hope it was not an unfavorable omen. "No," he replied,"it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."\par That is, $1729=1^{3} + 12^{3} = 9^{3}+10^{3}.$\par This anecdote reflects the extent of how well Ramanujan knew and studied the integers to have been aware and able to repeat this property while in the hospital.\par Generalizations of these numbers are now called \textit{taxicab numbers.}\newpage \section{Prime Numbers} A sample of a major result by Ramanujan is in his study of the prime numbers and is summarized in \cite{jonathan}.\par One of the most studied functions in number theory is the \textit{prime counting} function. For a real number $x$, we denote by $\pi(x)$ the number of prime numbers (i.e., those positive integers divisible by exactly two integers) less than or equal to $x$. So, for example, $\pi(10) =4$ because 2,3,5,7 are all the primes less than 10; it is left to the reader to verify that $\pi(100) = 25$. \begin{definition} For n $\geq$ 1, the $n-th$ \textit{Ramanujan prime} is the smallest positive integer $R_{n}$ with the property that if $x\geq n$, then $\pi(x)-\pi(\frac{1}{2}x) \geq n$.\par The sequence of Ramanujan primes is 2,11,17,29,41,.... A bound on the Ramanujan primes is given in the following theorem. \end{definition} \begin{theorem} The n-th Ramanujan prime satisfies the inequalities \begin{eqnarray} 2n log 2n \textless R_{n} \textless 4n log 4n (n \geq 1). \end{eqnarray} Furthermore,if $p_{n}$ denotes the $n-th$ prime, then $p_{2n} \textless R_{n} \textless p_{4n}, for \hspace{0.2cm} n \textgreater 1$. \end{theorem} \section{Partition Functions} Another area of number theory in which Ramanujan had worked extensively and is well-know for his contributions is the theory of \textit{partition functions}. The partition function $p(n)$ gives the number of ways of writing an integer $n$ as a sum of positive integers. Thus \ref{15} can be written in five different ways (without regard to order), as \begin{eqnarray} 4,3+1,2+2,2+1+1, \text{and} \hspace{0.3cm} 1+1+1+1. \end{eqnarray} The partition number $p(4)$ is therefore \ref{19}. The partition numbers of the integers 1 through 10 are given in Table \ref{1345}.\par \FloatBarrier \begin{table}[] \begin{tabular}{l|llllllllll} \hline \textbf{Number} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \textbf{p(n)} & 1 & 2 & 3 & 5 & 7 & 11 & 15 & 22 & 30 & 42 \\ \hline \end{tabular} \caption{The number of partitions for the integers 1 through 10.} \label{1345} \end{table} \FloatBarrier There is no formula that provides the values of $p(n)$ for given $n$, but the numbers can be determined through \textit{generating functions}. \section{Awards and Recognitions} Ramanujan has been recognized after his death with the following honors: \begin{enumerate} \item[$\bullet$]{Ramanujan's home estate (Tamil Nadu) celebrates Ramanujan's birthday (December 22) as a special holiday.} \item[$\bullet$]{The government of India in 1962 commemorated his life and achievements in number theory by issuing a stamp with his likeliness.} \item[$\bullet$]{A prize for young mathematicians was created in Ramanujan's name by the International Centre for Theoretical Physics and the International Mathematical Union.} \item[$\bullet$]{SASTRA University has established a prize of \$ 10000 to be awraded annually to a mathematician less than 32 years old for oustanding contributions in an area of mathematics influenced by Ramanujan. } \end{enumerate} \begin{thebibliography}{} \bibitem{Bruce} Bruce~C.~Berndt,\newblock {\em Ramanujan's Notebooks, Part III},\newblock { Springer-Verlng, New York }, 2002. \bibitem{Bruce1} Bruce~C.~Berndt,\newblock {\em Ramanujan's Notebooks, Part IV},\newblock { Springer-Verlng, New York }, 1994. \bibitem{george01} George~S.~Carr,\newblock {\em A Synopsis of Elemetary Results in Pure and Applied Mathematics},\newblock { (2 volumes) Cambridge University Press}, 2013. \bibitem{robert} Robert~Kanigel,\newblock {\em The Man Who Knew Infinity: A Life of the Genius Ramanujan},\newblock { Fifth edition, Washngton Square Press}, 1992. \bibitem{srinvasa} Srinivasa~Ramanujan,\newblock {\em Collected Papers},\newblock {American Mathematical Society}, 2000. \bibitem{jonathan} Jonathan~ Sondow,\newblock {Ramanujan Primes and Bertrand's Postulate},\newblock { \em The American Mathematical Monthly},Vol.~116, No.~7(Aug.-Sep.,2009),pp.630-635. \end{thebibliography} \end{document}