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\centerline{The Life and Work of Srinivasa Ramanujan}
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\beginsection Introduction
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Srinivasa Ramanujan (1887-1920) was a mathematician who made
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many significant contributions in the areas of hypergeometric series,
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divergent series, modular functions, the partition function,
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elliptic integrals and prime
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number theory.
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He was a prolific theorist and his results are still being studied
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today.
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\beginsection Outline
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\item{1.}
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Biography
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\item{2.}
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Work
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\itemitem{2.1} Divergent series
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\itemitem{2.2} A puzzle
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\itemitem{2.3} The partition function
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\itemitem{2.4} Highly composite numbers
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\itemitem{2.5} Continued fractions
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\itemitem{2.6} Rogers-Ramanujan Identity
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\itemitem{2.7} Selected formulas for $\pi$
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\itemitem{2.8} Ramanujan's tau function
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\itemitem{2.9} Other formulas
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\item{3.}
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References
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\beginsection 1. Biography
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In the winter of 1913, a distinguished mathematician in England
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received a letter from a clerk in India.
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The letter was ten pages long and was filled with handwritten
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mathematics.
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The English mathematician, Godfrey Hardy, showed the letter to his friends in the
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mathematics department at Cambridge.
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It caused quite a sensation.
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Many of the theorems in the letter, although unproved,
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were astonishing in their originality.
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It was generally agreed that the letter writer, S. Ramanujan,
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must be a mathematical genius.
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A month later, Hardy wrote back to Ramanujan.
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He praised Ramanujan's work and asked Ramanujan to send him
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proofs of the theorems.
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What Hardy did not put in the letter was the fact that he
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had already decided to bring Ramanujan to Cambridge.
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Back in India, the letter from the famous Mr. Hardy immediately
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boosted Ramanujan's career.
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On the basis of Hardy's letter,
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Ramanujan was able to secure a two year scholarship at the University of
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Madras and was able to quit his job as a shipping clerk.
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One thing that is important to know about Ramanujan is that, up until 1914, he
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had no formal training in advanced mathematics.
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He had taught himself mathematics by studying a book by Carr
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that he had aquired when he was fifteen years old.
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Ramanujan was twenty-five years old when he wrote to Hardy.
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He also wrote to H. F. Baker and E. W. Hobson.
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Hardy was the only one who replied.
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Ramanujan sent the letter because he had become somewhat desperate.
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He had attended college twice but did not finish because
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the only thing he wanted to study was mathematics.
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Unfortunately, his lack of a degree was holding him back.
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He was married and needed to work but was stuck as a shipping clerk.
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He dreamed of working as an academic so he could study
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mathematics full time.
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By writing to Hardy, he hoped to gain recognition in the mainstream mathematics
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community and thereby promote his career,
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which is in fact what happened.
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Despite not being a professional mathematician at the time,
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Ramanujan was actually well-known to mathematicians
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in Madras where he lived.
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In 1911 he published ``Some Properties of Bernoulli Numbers'' in the
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{\it Journal of the Indian Mathematical Society.}
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The paper was seventeen pages long.
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In the spring of 1914, at Hardy's behest, Ramanujan traveled by ship to England.
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He traveled alone and left his wife behind in India.
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It had taken Hardy a year to make all the arrangements.
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The plan was for Hardy and Ramanujan to work together at Cambridge University,
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and for Ramanujan to study for a degree.
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The book by Carr that Ramanujan had used to teach himself
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mathematics was about fifty years out of date.
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Ramanujan had to learn all of the new mathematics that had
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been discovered since then.
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In 1914, the year of his arrival in England, Ramanujan published one
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paper, ``Modular Equations and Approximations to Pi.''
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It was twenty-three pages long.
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In the following year (1915) he published nine papers.
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His most important paper that year was a fifty-two page treatise on highly composite numbers.
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This paper led to his award in 1916 of a Bachelor of Science by Research degree.
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Despite its title, this was actually an advanced degree.
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Also in 1916, Hardy and Ramanujan collaborated on a study of the partition function.
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The full account of their results was published in 1918.
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This was the most famous paper on which they collaborated because it
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established a new technology for attacking problems.
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Between the years 1915 and 1919, Hardy and Ramanujan collaborated
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on a total of four papers officially.
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In the spring of 1917, Ramanujan became sick with tuberculosis.
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He could not fully recover and he was in and out of hospitals over the next two years.
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In 1918 Ramanujan received three distinguished honors.
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He was elected Fellow of the Royal Society, Fellow of Trinity College, and Fellow of the
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Cambridge Philosophical Society.
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These honors boosted his spirts and he improved somewhat.
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However, he became sick again and decided, in 1919, to return to his
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home in India to recuperate.
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Unfortunately he did not recover and died in India in 1920 at the age of thirty-two.
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Throughout his life, Ramanujan accumulated all of his theorems in several large
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notebooks.
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When he arrived in England at the age of twenty-five, it is estimated that the notebooks
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already contained three to four thousand theorems.
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(Dr. Bruce C. Berndt, Professor of Mathematics at the University of Illinois,
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puts the figure at exactly $3,542$ theroems [5].)
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Since Ramanujan's death, the notebooks have taken on a life of their own.
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The majority of the results in the notebooks were unproven.
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Over the years, mathematicians have set out to prove Ramanujan's results and
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discovered new mathematics in the process.
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Hardy was in possession of the notebooks until 1928 when he gave them to G. N. Watson.
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Watson and B. M. Wilson ended up publishing over
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two dozen papers during the next ten years, all based on Ramanujan's original
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research.
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In 1977 the American mathematician Bruce C. Berndt took over the task.
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It took him one year to prove all of the results in just one of Ramanujan's chapters.
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As of 1991, Berndt was still at work investigating what Ramanujan had been working
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on more than seventy years earlier.
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%By 1940, over 100 papers based on Ramanujan's work had been published.
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\beginsection 2.1 Divergent series
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In his first letter to Hardy, Ramanujan included the following identity.
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$$1+2+3+\cdots=-{1\over12}$$
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Of course, this result cannot be obtained by summation.
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Rather, this result comes from two premises.
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First, if the sum of a divergent series is to have any meaning at all then the series
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itself must admit linear operations.
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Second, it is not admissible to reorder the terms of the series.
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This second premise guarantees the uniqueness of the result.
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At this point we need to establish an intermediate result. Let
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$$s=1-1+1-1+\cdots$$
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Then
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$$1-s=1-(1-1+1-1+\cdots)=1-1+1-1+\cdots=s$$
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Solving for $s$ we have $1=2s$ or $s=1/2$.
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Now let
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$$s=1+2+3+4+\cdots$$
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We are allowed to use linear operators so we can put
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$$-3s=s-4s$$
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Now $4s$ has an interesting property. We have $4s=2\cdot2s$ and so when subtracted
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from $s$ the result is that the signs of the even terms are flipped. Hence
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$$-3s=1-2+3-4+5-6+\cdots$$
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The right hand side can be written as the sum of two series as follows
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$$-3s=1-(1-2+3-4+\cdots)-(1-1+1-1+\cdots)=1+3s-1/2$$
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Solving for $s$ we have $-6s=1/2$ hence
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$$s=-{1\over12}$$
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from which it follows
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$$1+2+3+4+\cdots=-{1\over12}$$
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\beginsection 2.2 A puzzle
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In 1911, Ramanujan published the following puzzle in
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the {\it Journal of the Indian Mathematical Society.}
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The reader was asked to evaluate
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$$\sqrt{1+2\sqrt{1+3\sqrt{1+\cdots}}}$$
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After six months, no reader could solve the problem so Ramanujan
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provided the answer.
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$$\sqrt{1+2\sqrt{1+3\sqrt{1+\cdots}}}=3$$
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Ramanujan had crafted the puzzle from his theorem
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$$x+n+a=\sqrt{ax+(n+a)^2+x\sqrt{a(x+n)+(n+a)^2+(x+n)\sqrt{\ldots}}}$$
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In the puzzle, $x=2$, $n=1$, and $a=0$.
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\beginsection 2.3 The partition function
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Ramanujan spent a lot of time investigating the partition function.
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The partition function $p(n)$ is the number of ways the number
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$n$ can be expressed as the sum of addends.
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For example, $p(4)=5$ because there are five ways to partition 4.
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$$\eqalign{
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&4\cr
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&3+1\cr
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&2+2\cr
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&2+1+1\cr
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&1+1+1+1\cr
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}$$
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Note that the order of the addends is not significant.
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The addends are
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normally written from largest to smallest.
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Ramanujan discovered that the partition function has a $5k+4$
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congruence.
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In other words, if $n$ can be expressed as $5k+4$ (numbers that end in 4 or 9)
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then $p(n)$
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is divisible by 5.
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Ramanujan also discovered congruences for numbers that can be expressed
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as $7k+5$ and $11k+6$.
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Ramanujan and Hardy collaborated on a forty-page paper, published in 1918,
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that included a formula for computing the value of $p(n)$ from $n$.
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The formula was accurate up to $p(200)$.
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The exact formula for $p(n)$ was discovered twenty-one years later
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in 1937 by Hans Rademacher.
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\beginsection 2.4 Highly composite numbers
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In 1915 Ramanujan published a paper on highly composite numbers.
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Ramanujan himself invented the term ``highly composite.''
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A highly composite number is a number that has more factors than any
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number less than it. For example, the first six highly composite numbers are
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$$1,\quad2,\quad4,\quad6,\quad12,\quad24.$$
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The number 24 has eight factors.
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This is more factors than any number less than 24 has.
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While any composite number can be thought of as the ``opposite'' of a prime,
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a highly composite number is in a sense the most extreme opposite
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of a prime.
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(Except for 2 which is both prime {\it and} highly composite!)
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Ramanujan discovered that if you take a highly composite number
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and write down its prime factors from smallest to largest, then the
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powers of the primes {\it never} increase as you read from
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left to right. For example,
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$$24=2^3\times3^1$$
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In this case we have $3\ge1$.
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Ramanujan also discovered that, with the exception of 4 and 36,
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the final power is {\it always} 1, as it is in the above example.
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Ramanujan's published proof on this subject was fifty-two pages long.
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\beginsection 2.5 Continued fractions
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This formula involves the Golden Ratio $\phi$.
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$$\sum_{k=1}^\infty{1\over2^{\lfloor k\phi\rfloor}}={1\over2^0+\displaystyle{1\over2^1+\cdots}}$$
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\beginsection 2.6 Rogers-Ramanujan Identity
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$$1+\sum_{k=1}^\infty{q^{k^2+k}\over(1-q)(1-q^2)\cdots(1-q^k)}
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=\prod_{j=0}^\infty{1\over(1-q^{5j+2})(1-q^{5j+3})},\qquad|q|<1$$
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\beginsection 2.7 Selected Formulas for $\pi$
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The following Ramanujan formula for $\pi$ is accurate to eight decimal places.
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$$\pi\approx\left(97+{1\over2}-{1\over11}\right)^{1/4}$$
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The following Ramanujan formula was used to calculate $\pi$ to one billion digits in 1989.
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$${1\over\pi}={\sqrt8\over9801}\sum_{n=0}^\infty
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{(4n)!(1103+26{,}309n)\over(n!)^4(396)^{4n}}$$
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\beginsection 2.8 Ramanujan's tau function
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$$\sum_{n=1}^\infty\tau(n)x^n=x\prod_{n=1}^\infty(1-x^n)^{24}$$
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The coefficients for $\tau$ can be calculated as follows.
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$$\sum_{n=1}^\infty\tau(n)x^n=x(1-3x+5x^3-7x^6+9x^{10}-\cdots)^8$$
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\beginsection 2.9 Other formulas
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In 1916 Ramanujan published ``On certain arithmetical functions'' that contained
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many identities involving the functions
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$$L(q)=1-24\sum_{n=1}^\infty{nq^n\over1-q^n},\quad
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M(q)=1+240\sum_{n=1}^\infty{n^3q^n\over1-q^n},\quad
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N(q)=1-504\sum_{n=1}^\infty{n^5q^n\over1-q^n}$$
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which are defined for $|q|<1$.
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A few of the identities are
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$$
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q{dL\over dq}={L^2-M\over12},\qquad
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q{dM\over dq}={LM-N\over3},\qquad
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q{dN\over dq}={LN-M^2\over2}.
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$$
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These identities are applicable to the series
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$$\sum_{m,n=-\infty}^\infty q^{m^2+mn+2n^2}$$
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%\beginsection 2.4 Other formulas
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\noindent
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This formula equates the infinite sum on the left with a finite sum on the right.
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$$\sum_{k=1}^\infty{1\over n+k}={n\over2n+1}+\sum_{k=1}^n{1\over(2k)^3-2k}$$
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\beginsection 3. References
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\item{1.}
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Kanigel, Robert. {\it The Man Who Knew Infinity.} New York: Washington Square Press,
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1991.
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\item{2.}
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Chan and Ong. {\it On Eisenstein Series.}
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{\tt http://www.ams.org/proc/\hfill\break
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1999-127-06/S0002-9939-99-04832-7/S0002-9939-99-04832-7.pdf}
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\item{3.}
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Cais, Bryden. {\it Divergent Series.}\hfill\break
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{\tt http://www.math.uiuc.edu/$\sim$berndt/articles/rrcf.pdf}
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\item{4.}
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{\it Srinivasa Ramanujan.} Wikipedia.\hfill\break
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{\tt http://en.wikipedia.org/wiki/Srinivasa\_Ramanujan}
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\item{5.}
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{\it Ramanujan's Tau Function.}\hfill\break
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{\tt http://www.users.globalnet.co.uk/\hfill\break
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$\sim$perry/maths/ramanujantau/ramanujantau.htm}
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\item{6.}
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{\it Rediscovering Ramanujan.} Frontline.
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{\tt http://www.hinduonnet.com/\hfill\break
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fline/fl1617/16170810.htm}
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\end
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