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.
The second shortcut is more cunning. The cotangent has a twin, the hyperbolic cotangent, which you get by feeding the same function an imaginary angle. When you add the two, every second layer cancels out, the way two waves can cancel each other; when you subtract them, the other half cancels.
For the bottoms he finds a rule so simple it could be a pencil game. You take the index of the Bernoulli number, list everything that divides it, add one to each, keep only the primes, and multiply them: that is the bottom of the fraction. It matches every entry in his table.
Where do the rough limits come from in the first place? Here he borrows a classical bridge. There is a famous sum: add one over every whole number squared, or cubed, or raised to any even power, and you get a definite number; for squares it is pi squared divided by six.
His second instrument compares neighbours. Each Bernoulli number turns out to be very nearly the one two places before it, multiplied by a simple factor built from the index and pi. The approximation becomes excellent very quickly as the numbers grow.
To show the recipe working, he carries out the five-interval case. The five fifth roots of one sit at the corners of a regular pentagon, and the pentagon brings in the golden ratio, the number about one point six one eight famous from sunflowers, seashells and the Fibonacci numbers.
Why read such a paper today? Because it is a rare look at a great mind before it was trained: a young man working alone in Madras, rediscovering a famous theorem, inventing fast ways to compute, 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.
The lectures are free to read, share and quote with a link to this page.
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