#### (1887-1920)

# Srinivasa Ramanujan

## An equation for me has no meaning, unless it expresses a thought of God.”- Srinivasa Ramanujan

Srinivasa Ramanujan, often referred to simply as Ramanujan, was a brilliant Indian mathematician who made extraordinary contributions to the field of mathematics in the early 20th century. Despite his humble background and lack of formal training, Ramanujan’s innate mathematical talent and insights led to groundbreaking discoveries in areas like number theory, infinite series, and modular forms. His work continues to be a source of inspiration for mathematicians and scientists worldwide, and he is considered one of the greatest mathematical minds in history.

## The Formative Years of Srinivasa Ramanujan

Srinivasa Ramanujan, a fine sensation, was born in the Tamil Nadu state’s small city of Erode on December 22, 1887. Since he was born into a Tamil Brahmin Iyengar family, his early life was marked by humble onsets. While his mother, Komalatammal, offered her beautiful voice to the neighborhood temple, his father, Kuppuswamy Srinivasa Iyengar, made a humble wage as a sari shop clerk.From the young Ramanujan’s journey was anything from conventional. His life tragically changed at the age of barely one and a half when his infant brother, Sadagopan, lost suddenly.Soon after, in 1889, he battled a severe bout of smallpox, surviving against the odds when many did not. Following these trials, he and his mother moved to Kanchipuram, near Madras, seeking stability.The years that followed were marked by both hardships and the nurturing of Ramanujan’s extraordinary talents. Despite facing numerous setbacks, including the

loss of two more siblings, Ramanujan’s intellectual prowess began to shine through. He enrolled in school in 1892, marking the start of his formal education. His journey through various schools eventually led him to Town Higher Secondary School, where he encountered formal mathematics for the first time.

Remarkably, by the age of 11, Ramanujan had already surpassed the mathematical knowledge of college students. His voracious appetite for learning led him to devour books on advanced trigonometry and develop complex theorems independently. Merit certificates and academic awards adorned his school years, and his grasp of mathematics far exceeded his peers.

In 1904, as he graduated from Town Higher Secondary School, Ramanujan received the K. Ranganatha Rao prize for mathematics, a testament to his exceptional abilities. However, his pursuit of mathematics became all-consuming, leading to academic struggles in other subjects and the loss of a scholarship.

Undeterred, Ramanujan embarked on a path of independent mathematical research.Living in hardship and frequently on the verge of famine, he persisted in his studies and the development of his mathematical prowess. In 1910, when he met V. Ramaswamy Aiyer, the creator of the Indian Mathematical Society, his life underwent a tremendous change. This meeting marked the beginning of recognition for the brilliant mathematician within the mathematical circles of Madras.

Ramanujan’s early life was marked by adversity, but it was also the crucible in which his exceptional mathematical talent was forged, eventually earning him a place among the greatest mathematicians in history.

## Mathematical Pursuits after meeting V. Ramaswamy Aiyer

After meeting Deputy Collector V. Ramaswamy Aiyer in 1910 and showing him his mathematics notebooks, Srinivasa Ramanujan’s career in mathematics began to gain momentum. Aiyer was deeply impressed by the extraordinary mathematical results within Ramanujan’s notebooks and sought to support his talents. Aiyer introduced Ramanujan to mathematicians in Madras, including R. Ramachandra Rao, the Indian Mathematical Society’s secretary and the district collector for Nellore. Originally, there were doubts about whether Ramanujan’s work was truly his own, but Ramanujan’s continuity and conversations about his correspondence with Professor Saldhana helped palliate these enterprises.

With fiscal support from Rao and stimulants from Aiyer, Ramanujan had his work published in the Journal of the Indian Mathematical Society. His published work covered different fine motifs, including elliptic integrals, hypergeometric series, and divergent series. Notably, Ramanujan presented an unsolved problem in the journal, which he later provided an incomplete solution to. He similarly excavated into the plots of Bernoulli figures, discovering that the denominators of Bernoulli fragments were always separable by six.

In 1912, Ramanujan secured a temporary job in the Madras Accountant General’s office, but his passion for mathematics led him to spend utmost of his time on independent fine exploration. He latterly applied for a position under the Chief Accountant of the Madras Port Trust and was accepted.

In 1913, Ramanujan’s work caught the attention of British mathematicianG.H. Hardy, who originally viewed Ramanujan’s calligraphies with dubitation . Still, upon near examination, Hardy honored the genius behind Ramanujan’s theorems and formulas. Hardy and Ramanujan began a correspondence that ultimately led to Ramanujan traveling to England to unite with Hardy at the University of Cambridge. This marked the morning of Ramanujan’s career in the transnational fine community, where he made significant contributions to the field of number propositions and continued to publish groundbreaking work.

### Ramanujan's Equation-Filled Odyssey: From India to England

Upon his arrival in London on April 14, 1914, Srinivasa Ramanujan was warmly greeted by E. H. Neville, who had a car waiting for him. Neville then transported him to his residence on Chesterton Road in Cambridge. Here, Ramanujan commenced his collaborative work with G. H. Hardy and J. E. Littlewood. After a brief six-week stay at Neville’s, he relocated to Whewell’s Court, situated just a stone’s throw away from Hardy’s quarters.

#### The Mathematicians' Encounter

- Ramanujan, Hardy and Littlewood began delving into Ramanujan’s extensive notebooks. In the initial two letters alone, Ramanujan had shared 120 theorems, yet his notebooks contained an abundance of additional results and theorems. Hardy soon realized that some of these were incorrect, some had already been established by other mathematicians, while others represented groundbreaking discoveries.

#### A Unique Collaboration: Hardy and Ramanujan

- Ramanujan spent nearly five fruitful years in Cambridge, working closely with Hardy and Littlewood and publishing significant portions of his findings during this period. Their collaboration was an intriguing mix of clashing personalities and working styles. While Hardy championed mathematical rigor and proof, Ramanujan, a devoutly religious man, relied heavily on intuition and insight. Hardy made efforts to bridge the gaps in Ramanujan’s formal education, emphasizing the importance of rigorous proofs without stifling his remarkable creativity. This dynamic gave rise to a truly unique mathematical partnership.

#### Recognition and Achievements

- In recognition of his groundbreaking work on highly composite numbers, Ramanujan was awarded a Bachelor of Arts by Research degree in March 1916. This marked a significant achievement, with his research showcasing his exceptional grasp of the algebra of inequalities, even in what some considered the “backwater of mathematics.”

## Ramanujan's prominence kept growing. He was tagged to the London Mathematical Society in December 1917, and in May 1918, he was named a Fellow of the Royal Society, one of the loftiest honors in the discipline. He was one of the youthful people in the history of the society to achieve this coveted rank at the age of just 31.His election honored his significant contributions to the Theory of Numbers and elliptic functions. In addition, he made history on October 13, 1918, when he became the first Indian to be elected a Fellow of Trinity College, Cambridge.

### Mathematical Marvel: Contributions That Transformed the Field

In the realm of mathematics, Ramanujan’s contributions spanned beyond traditional proofs and theorems; they embodied profound mathematical insights and a wealth of unexplored formulae. G. H. Hardy recognized the depth of Ramanujan’s discoveries, often revealing hidden layers beneath the surface. Ramanujan’s work paved the way for new avenues of research.

#### A Formula for π's Swiftness

- One of Ramanujan’s most remarkable achievements was the creation of an infinite series for π. This formula converges with astonishing rapidity and serves as the foundation for some of today’s fastest π-calculation algorithms. By truncating the series after the first term, an approximation for π accurate to six decimal places is obtained, while two terms yield a remarkable 14 decimal places of accuracy. This work also laid the groundwork for the broader Ramanujan–Sato series

#### Problem-Solving Prowess

- Ramanujan possessed an uncanny ability to swiftly solve complex mathematical problems. A notable anecdote involves P. C. Mahalanobis presents him with a challenging problem about house numbering. Ramanujan’s response was not a conventional solution but a continued fraction—a solution encompassing a whole class of problems. His intuitive insights transcended traditional methods, leading him to derive previously unknown identities.

#### A Mathematical Revelation

- Ramanujan’s intuition led him to unveil hidden mathematical identities, such as a pair of intricate equations involving trigonometric and hyperbolic functions. These revelations, closely tied to the gamma function and the Dedekind eta function, opened doors to new avenues of exploration within the realm of mathematics.

#### The Partition Function and the Circle Method

- In collaboration with Hardy, Ramanujan delved deeply into the partition function P(n). Their work yielded a non-convergent asymptotic series, enabling precise computation of integer partitions. Later refined by Hans Rademacher, this formula evolved into a convergent series solution. Their pioneering efforts laid the foundation for the circle method, a potent tool for deriving asymptotic formulae.

#### Mock Theta Functions

- In the final year of his life, Ramanujan unearthed mock theta functions—a mathematical enigma. While they remained a mystery for some time, these functions are now recognized as the holomorphic components of harmonic weak Maass forms, adding another layer to Ramanujan’s enduring legacy in mathematics.

## The noteworthy accomplishments of Srinivasa Ramanujan

Srinivasa Ramanujan’s luminous achievements include his remarkable fine prowess, as substantiated by his early mastery of advanced textbooks, similar to Loney’s workshop, when he was just 12 years old. In 1916, he earned a Bachelorette of Science degree” by exploration” from Cambridge University, a testament to his fine benefactions

His exceptional gift was further conceded in 1918 when he became the first Indian to be tagged as a Fellow of the Royal Society. In 1997, The Ramanujan Journal was established to publish exploration told by his work.

To recognize his enduring heritage, the year 2012 was designated as the National Mathematical Year, marking the 125th anniversary of his birth. Also, since 2021, December 22, his birth anniversary, has been celebrated as National Mathematicians Day in India. These enterprises aim to inspire and cultivate a deep interest in mathematics among the country’s youth, feting their part in shaping the nation’s future.