Debunking Ramanujan

International Journal for Research in Applied Science & Engineering Technology (IJRASET), 2022

As we know that Sir Ramanujan gave the solution of sum of all natural numbers up to infinity and said that the sum of all natural numbers till infinity is-1/12. I studied on this topic and found that if we try to solve the infinite series in a slightly different way, then we get the answer of its sum different from-1/12, so this is what I have written in this paper that such Ramanujan Sir, what was the mistake in solving the infinite series, which by solving it in a slightly different way from the same concept, we get different answers.

arXiv: Number Theory, 2016

We give a new appraisal of a famous oscillating power series considered by Hardy and Ramanujan related to the erroneous theory of distribution of primes by Ramanujan.

Srinivasa Ramanujan, one of greatest mathematicians in history, made path-breaking contributions without any formal education, and undeterred by poverty and hardships. Ramanujan's accomplishments are compared with those of certain mathematical luminaries who in their own way faced tremendous difficulties in life, yet made revolutionary contributuions. =============== 2. Ramanujan: The second century Abstract: Srinivasa Ramanujan's spectacular discoveries revealed surprising connections between apparently unrelated topics, and provided food for thought for mathematicians of subsequent generations. This article describes how Ramanujan's results and ideas will continue to influence research in the century following his centenary in areas such as mock theta functions, congruences for partition functions and coefficients of modular forms, qhypergeometric identities, special functions, mathematical physics, and computer algebra. =============== 3. L. J. Rogers: A contemporary of Ramanujan Abstract: The British mathematician L. J. Rogers had talents similar to Ramanujan in the theory of q-hypergeometric series and had discovered and proved the Rogers-Ramaujan identities about two decades before Ramanujan discovered them. In this article the life and mathematical contributions of Rogers are described and the story told of how Ramanujan in England accidentally came across certain papers of Rogers, and the recognition Rogers received after Ramanujan's rediscovery of his work.

Ramanujan's story is one of the great romantic tales of mathematics. It is an account of triumph and tragedy, of a man of genius who prevailed against incredible adversity and whose life was cut short at the height of his powers. The extent of those powers is only now being fully recognized. Ramanujan had the misfortune to work on problems that, in his time, were considered a mathematical backwater. Modular equations, theta function identities, even continued fractions were viewed as having been played out in the nineteenth century. One might pick up tidbits, but there was nothing important left to be discovered. G.H. Hardy knew the error of this view. In his twelve lectures given at Harvard in 1936 [31], he communicated the range and depth of Ramanujan's work. Their asymptotic series for the number of partitions of an integer, published in 1918 [32], later refined by Rademacher [35] into a rapidly convergent series, remains one of the great achievements of analytic number theory. Hardy credited Ramanujan for all of the inspiration. Nevertheless, even Hardy expressed uncertainty about the true greatness of Ramanujan's accomplishments:

The essay component here is to investigate the 'Dual-Pillars' of mathematics; while the review is motivated by two of my lifetime obsessions: physics and philosophy, so I looked forward to reading this book after seeing the film of the same title (released in 2015 – 24 years after the book), starring Dev Patel and Jeremy Irons. That enjoyable film encouraged me to find out more about the Cambridge mathematician, G. H. Hardy, so I read his famous book " A Mathematician's Apology " and wrote a critical essay thereon, as it appeared more like a justification for fellow " pure " mathematicians than an informative autobiography. This review now 'balances the books' for me, by herein writing a book about the true genius in this famous duo – the self-educated Indian, S. Ramanujan. This book adds invaluable insights into this incredible man's early life, whereas the film only begins as Ramanujan gets his first job at 25 in Madras <book=p.95>; in reality, Ramanujan made a huge personal effort to make others aware of his major innovations caused by his total obsession with mathematics causing him to lose scholarships and a degree that all required he have a 'rounded' education. Furthermore, the author visited South India for book research, so providing an in-depth cultural background to illustrate the spiritual forces, inspiring and motivating Ramanujan. In particular, an informative discussion of the Hindu caste system, especially the top caste of Brahmins (even impoverished ones, such as Ramanujan). If the readers of this review have ever been touched by the 'Magic of Math' (as I have) then they will really enjoy this book, even if they saw the film first (as I did). Those who were never touched by this 'magic' (most of the population) will still gain great insights into any personality, who has lived in the grip of any abstract intellectual obsession. The drama in this highly personalized story is the conflict between the conception of mathematics in the Western tradition, following the classic Greeks and their obsession with geometry and the use of " proofs " to demonstrate the logic <222> of the results. In contrast, the Indian tradition emphasized intuition and 'getting the results'; this was Ramanujan's natural mode for discovering his brilliant results. Ironically, the Emperor of Intuition was discovered by the Prince of Proof. But professional mathematicians also want to see a 'proof' so they might follow the developmental logic as very few have the power of intuitive insight to see the deep relationships. While the West was still struggling with Roman numerals (e.g. VII=7), the ancient Indian scholar Bhaskara was the first around 630 CE to write numbers in the Hindu decimal system with a new sign for zero. Although stimulated by studies of the stars (like many societies) to define their calendar, the Indians were already investigating algebra, geometry and trigonometry – the main areas of modern universal mathematical education. Indeed, one of their greatest scholars, Aryabhata wrote his masterpiece (later called by his successors, the Aryabhatiya) about 500 CE, which has survived to modern times (but was not studied by Ramanujan). In addition to much useful astronomical work, this mathematics covered arithmetic, algebra, plane and spherical trigonometry; it also included continued-fractions, quadratic equations, sums of power series and a table of sine values. It was written in the form of tight poetry (to aid in its memorization) in 108 verses following 13 introductory verses. The explication of its meaning is due to later commentators. Ironically, Ramanujan's Indian style of mathematics was encouraged by his first encounter (when 16) with a formal text book: A Synopsis of Elementary Results in Pure and Applied Mathematics by Englishman, George Carr. That book was simply a compilation of over 5,000 equations that Carr used as a professional tutor for the infamous mathematics Examination at Cambridge University known as the Tripos. It was this style of mass memorization that has given mathematics a deservedly bad name; however, a good memory is still vital to any success in mathematics: if not just equations, then for remembering all the definitions, rules and methods. Ramanujan did not simply remember all Carr's formulas but worked out (or 'proved' each one to himself) as activity is also necessary to master the extensive subject of mathematics.

Archiv der Mathematik, 1995

Mathematics

The main contribution of this paper is to propose a closed expression for the Ramanujan constant of alternating series, based on the Euler–Boole summation formula. Such an expression is not present in the literature. We also highlight the only choice for the parameter a in the formula proposed by Hardy for a series of positive terms, so the value obtained as the Ramanujan constant agrees with other summation methods for divergent series. Additionally, we derive the closed-formula for the Ramanujan constant of a series with the parameter chosen, under a natural interpretation of the integral term in the Euler–Maclaurin summation formula. Finally, we present several examples of the Ramanujan constant of divergent series.

Bulletin of the American Mathematical Society, 2006

Ramanujan's story is one of the great romantic tales of mathematics. It is an account of triumph and tragedy, of a man of genius who prevailed against incredible adversity and whose life was cut short at the height of his powers. The extent of those powers is only now being fully recognized. Ramanujan had the misfortune to work on problems that, in his time, were considered a mathematical backwater. Modular equations, theta function identities, even continued fractions were viewed as having been played out in the nineteenth century. One might pick up tidbits, but there was nothing important left to be discovered. G.H. Hardy knew the error of this view. In his twelve lectures given at Harvard in 1936 [31], he communicated the range and depth of Ramanujan's work. Their asymptotic series for the number of partitions of an integer, published in 1918 [32], later refined by Rademacher [35] into a rapidly convergent series, remains one of the great achievements of analytic number theory. Hardy credited Ramanujan for all of the inspiration. Nevertheless, even Hardy expressed uncertainty about the true greatness of Ramanujan's accomplishments:

Solved and unsolved mathematical problems (Latest Edition: January 2020), 2020

The Ramanujan Journal, 2012

We give a natural derivation of a formula of Ramanujan, described by B.C. Berndt as "enigmatic", for the harmonic series.