Collected Papers of Srinivasa Ramanujan
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The first paper the self-taught genius ever published, written in Madras at 23: strange patterns in the Bernoulli numbers, the numbers behind sums of powers. Most of his claims are true, some are guesses, and one is false; the lectures show which is which.
This is a sample: 1 of 37 papers.
That function is the cotangent from school trigonometry, multiplied by the angle. Written out near zero as a long sum of powers, its successive terms carry the Bernoulli numbers one after another; Ramanujan takes them all as positive and calls the zeroth one minus one so that the first term fits.
A close cousin of the rule says that the part of each Bernoulli number after its whole part is made of one over each of those primes, added up, minus one.
Then he tries to prove his own patterns. His plan is to show that certain infinite sums, built from one over the squares of shifted whole numbers and from the primes above three, can be rewritten as series whose coefficients are all whole numbers.
Third, the showpiece: the Bernoulli number with index four hundred and forty-four. In 1911 writing it out by hand was out of the question, yet Ramanujan describes it in detail.
To show it working, he carries out the five-interval case. The fifth roots of one bring in the golden ratio, the number about one point six one eight that appears in sunflowers, in the Fibonacci numbers and in classical architecture.
Why is this still worth reading? Because it shows a mathematical mind at the very start: self-taught, without access to the literature, rediscovering a famous theorem, finding fast computational shortcuts, and guessing patterns from data with a sure instinct for which ones matter.
Examples for: a curious adult with school maths who likes number puzzles and knows Ramanujan's story; add one modern instance per idea, such as a calculator, computer algebra or cryptography
The lectures retell the English text of the Cambridge 1927 edition (Internet Archive pli.kerala.rare.28155). The author died in 1920; the work is in the public domain. The retelling, the drawings and the voice are made by AI and checked against the book by a second model; nothing was edited by hand. How AI is used.
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