Ramanujan’s most beautiful identity

G. H. Hardy called the following equation Ramanujan’s “most beautiful identity.” For |q| < 1,

\sum_{n=0}^\infty p(5n+4) q^n = 5 \prod_{n=1}^\infty \frac{(1 - q^{5n})^5}{(1 - q^n)^6}

If I understood it, I might say it’s beautiful, but for now I can only say it’s mysterious. Still, I explain what I can.

The function p on the left side is the partition function. For a positive integer argument n, p(n) is the number of ways one can write n as the sum of a non-decreasing sequence of positive integers.

The right side of the equation is an example of a q-series. Strictly speaking it’s a product, not a series, but it’s the kind of thing that goes under the general heading of q-series.

I hardly know anything about q-series, and they don’t seem very motivated. However, I keep running into them in unexpected places. They seem to be a common thread running through several things I’m vaguely familiar with and would like to understand better.

As mysterious as Ramanujan’s identity is, it’s not entirely unprecedented. In the eighteenth century, Euler proved that the generating function for partition numbers is a q-product:

\sum_{n=0}^\infty p(n) q^n = \prod_{n=1}^\infty \frac{1}{(1 - q^n)}

So in discovering his most beautiful identity (and others) Ramanujan followed in Euler’s footsteps.

Reference: An Invitation to q-series

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6 thoughts on “Ramanujan’s most beautiful identity

  1. Given his unorthodox background, I wonder if Ramanujan knew of Euler’s work. In any case, I am constantly amazed of these things that “real” mathematicians dream up!

  2. Ramanujan wasn’t entirely self-taught. Hardy mentions some books as ones Ramanujan had definitely read, others he was unsure about.

    It’s plausible that Ramanujan rediscovered Euler’s identity. He was interested in partitions, and it’s natural to want to find the generating function of a sequence you’re interested in.

  3. shashikant shukla

    we need to remember that Ramanujan was just 32 years when he died, was much more poor than general definition of poor known in western countries, was working as full-time clerk (note this is in India), and stayed in UK just for 5 years (1914 to 1919)…. he neither had much time, nor money or means to get books, nor living in internet era.. so probably it was in UK where he would get to read anything on partitions.. or any other problems he was working on in UK

    if he would have lived even 50 years, his contributions would be much more dramatic..

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