Indian mathematician's genius formula from a century ago might explain the dark secrets of black holes

It was in the year 1914 that Indian mathematician Srinivasa Ramanujan came to Cambridge with a notebook filled with 17 extraordinary infinite series for 1/π. They were not only efficient but also gave accurate digits of the world's most famous irrational number.

And even though for a century, the formulas were viewed as pinnacle of number theory, no one could actually explain how they worked so perfectly. However, Now, researchers at the Indian Institute of Science (IISc) have discovered an unexpected link between Ramanujan’s famous formulas for pi and the modern physics used to explain black holes and turbulent fluids. The study suggests that Ramanujan was, perhaps unknowingly, working with the same underlying mathematics that describes how matter behaves at the brink of extreme change.

Ramanujan’s remarkable pi formulae

In 1914, just before leaving Madras for Cambridge, Indian mathematician Srinivasa Ramanujan published a paper that listed 17 different formulas for calculating pi. These formulas stood out because they were far more efficient than the methods available at the time. With only a few mathematical terms, they could generate a surprisingly large number of correct digits of pi.More than a century later, their influence remains strong.

Ramanujan’s ideas form the basis of modern techniques used to compute pi on powerful computers today. “Scientists have computed pi up to 200 trillion digits using an algorithm called the Chudnovsky algorithm,” says Aninda Sinha, professor at the Centre for High Energy Physics (CHEP) and senior author of the study. “These algorithms are actually based on Ramanujan’s work.

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A deeper question behind the mathematics

For Sinha and Faizan Bhat, the study’s first author and a former PhD student at the Indian Institute of Science (IISc), the interest was not just in how fast these formulas work.

They wanted to understand why such powerful formulas exist at all. Instead of viewing them as purely abstract mathematics, the researchers looked for a connection to the physical world.“We wanted to see whether the starting point of his formulae fit naturally into some physics,” says Sinha. “In other words, is there a physical world where Ramanujan’s mathematics appears on its own?”

Where pi meets scale invariance and extreme physics

Their search led them to a class of theories known as conformal field theories, and in particular to logarithmic conformal field theories.

These theories describe systems with scale invariance, meaning the system looks the same no matter how much you zoom in or out.A well-known example is water at its critical point, a specific temperature and pressure where liquid water and water vapour become indistinguishable. At this point, water shows scale-invariant behaviour, which can be described using conformal field theory. Similar behaviour appears in processes such as percolation, the early stages of turbulence in fluids, and in certain theoretical descriptions of black holes.

These are all areas where logarithmic conformal field theories are used.

Using Ramanujan’s structure to tackle physics problems

The researchers found that the same mathematical structure underlying Ramanujan’s pi formulas also shows up in the equations of these logarithmic conformal field theories. By using this shared structure, they were able to calculate important quantities in these theories more efficiently. This could help scientists better study complex phenomena like turbulence and percolation.This approach closely reflects Ramanujan’s own style, where compact mathematical expressions lead quickly to precise results. “[In] any piece of beautiful mathematics, you almost always find that there is a physical system which actually mirrors the mathematics,” says Bhat. “Ramanujan’s motivation might have been very mathematical, but without his knowledge, he was also studying black holes, turbulence, percolation, all sorts of things.

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A century-old insight with modern impact

The study shows that Ramanujan’s work from over 100 years ago still offers new tools for making modern high-energy physics calculations faster and easier. Beyond the technical benefits, the researchers say the findings underline the extraordinary depth of his ideas.“We were simply fascinated by the way a genius working in early 20th century India, with almost no contact with modern physics, anticipated structures that are now central to our understanding of the universe,” says Sinha.