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Collected Papers ofSrinivasa Ramanujan

Srinivasa Ramanujan · 1911 · retold from the English original

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.

From the lectures

Rows of coefficient boxes for the cotangent and its hyperbolic twin; adding them leaves only every second Bernoulli number, and a bar half as long shows the reduced work.
A Family of Fractions in the Cotangent

The twin cancels every second term

His second improvement mixes the ordinary cotangent with its hyperbolic twin, which is the same function evaluated on an imaginary argument.

A bar chart of the size of the first twenty Bernoulli numbers on a power-of-ten scale: small at first, then climbing to about 10 to the 16.
Twenty Fractions and the Patterns in Them

Twenty specimens

Going on like this, he lists twenty Bernoulli numbers. They stay small for a while, one sixth, one thirtieth, one forty-second, one thirtieth again, five sixty-sixths, then they climb fast.

The formula: a Bernoulli number is two times n factorial over two pi to the n, times a sum just above one; for index two it gives exactly one sixth; for ten factorial the logarithm expansion gives 15.10441.
Pinning a Number From Its Size

Size from a factorial

Where do such rough limits come from? Section eleven quotes a classical formula: a Bernoulli number equals two, times the factorial of its index, divided by two pi raised to the index, times the sum of one over every whole number raised to that index.

Three routes, the reciprocal squares, the alternating tail and the harmonic sum, each lead to a result only through an unproved bridge: if we can prove.
A Route to Proof, With One Gap

Three routes, one missing bridge

He then repeats the trick twice. Taking the same tail with alternating signs, he gets a series whose coefficients are Bernoulli numbers multiplied by one less than a power of two; if twice that alternating tail had whole-number coefficients, the second half of his oddness statement would follow.

One-, two- and three-interval formulas; p-th roots of one (or of i for even p) mixed into the cotangent give p-interval formulas with strands spaced 2p apart; the five fifth roots of one at the corners of a regular pentagon; the sum of sin xα over the five roots keeps only every tenth power: 5(x⁵/5! − x¹⁵/15! + ⋯).
Other Starting Points, and Five-Way Sieves

Five roots on a pentagon

Second, the number of ways to cancel. Ramanujan names his earlier results the one-interval, two-interval and three-interval formulas, after how far apart the surviving Bernoulli numbers sit.

The editors' note: different numbering and binomial coefficients; the recurrence holds for odd indices above one; two derivations invalid because a series diverges; the main results known (von Staudt–Clausen, Euler); of eight statements, three have conditional proofs, two are consequences, three are conjectures.
The Four-Way Sieve and the Final Verdict

Three kinds of claim

The book then prints a note from the editors of the collected papers, and it is the right place to take stock. First, small clarifications: Ramanujan numbers his Bernoulli numbers differently from most books, and the counting numbers in his equations are the usual binomial coefficients.

How it was made

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 source

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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