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.
Matching coefficients as before gives equation eight, for even n greater than 2: c₂Bₙ₋₂/2 − c₆Bₙ₋₆/2³ + c₁₀Bₙ₋₁₀/2⁵ − ⋯ + (n/2^((n+2)/2))(−1)^(n/4) = 0, the sign being (−1)^(n/4) when n is a multiple of 4 and (−1)^((n−2)/4) when n minus 2 is.
Carrying on this way, he lists twenty values, from B₂ = 1/6 to B₄₀. They stay small at first, B₁₄ = 7/6, then climb fast: B₂₀ = 174611/330, about 529, and B₄₀ has a numerator of 21 digits over a denominator of only 13530.
Section ten puts these together. Suppose all you know is that B₂₂ lies between 6084 and 6244. Its fractional part must be 17/138, its numerator must be divisible by 11, and B₂₂ − 11/6 must be divisible by 5. Among the 160 whole numbers in that range only two survive: 6137 17/138 and 6192 17/138.
Section fourteen compares neighbours. Dividing equation twenty-four for n by the same for n minus 2 gives equation thirty-four: Bₙ = (n(n − 1)/(4π²)) Bₙ₋₂ (1 − (2² − 1)/(2ⁿ − 1))(1 − (3² − 1)/(3ⁿ − 1))⋯.
Section sixteen notes that the tangent and the cosecant can replace the cotangent. One result is equation thirty-six: (4/3)Bₙ(2ⁿ − 1) = c₆Bₙ₋₆(2ⁿ⁻⁶ − 1) − c₁₂Bₙ₋₁₂(2ⁿ⁻¹² − 1) + ⋯, plus a last term ⅓n or ⅙n with a sign. For n equal to 6: (4/3) × (1/42) × 63 = 2, and the last term is ⅓ × 6 = 2.
The book then adds a note from the editors of the collected papers, and it is worth reading closely, because it sorts everything we have met by its status. First, small clarifications: Ramanujan writes B₂ₙ where most books write Bₙ, and cₖ is the binomial coefficient n!/(k!(n − k)!).
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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