Decimal Digits of 1/π²

This formula may let you compute a decimal digit of 1/\pi^2 without computing all the previous digits:

\displaystyle{ \frac{1}{\pi^2} = \frac{2^5}{3} \sum_{n=0}^\infty \frac{(6n)!}{n!^6} (532n^2 + 126n + 9) \frac{1}{1000^{2n+1}}  }

It was discovered here:

• Gert Almkvist and Jesús Guillera, Ramanujan-like series for 1/\pi^2 and string theory, Experimental Mathematics, 21 (2012), 223–234.

They give some sort of argument for it, but apparently not a rigorous proof. Experts seem to believe it:

• Tito Piezas III, A compilation of Ramanujan-type formulas for 1/\pi^m.

It’s reminiscent of the famous Bailey–Borwein–Plouffe formula for \pi:

\displaystyle{ \pi = \sum_{n = 0}^\infty  \frac{1}{16^n} \left( \frac{4}{8n + 1} - \frac{2}{8n + 4} - \frac{1}{8n + 5} - \frac{1}{8n + 6} \right)  }

This lets you compute the nth hexadecimal digit of \pi without computing all the previous ones. It takes cleverness to do this, due to all those fractions.

A similar formula was found by Bellard:

\begin{array}{ccl}  \pi &=& \displaystyle{ \frac{1}{2^6} \sum_{n=0}^\infty \frac{(-1)^n}{2^{10n}} \, \left(-\frac{2^5}{4n+1} - \frac{1}{4n+3} +  \right. } \\ \\  & & \displaystyle{ \left. \frac{2^8}{10n+1} - \frac{2^6}{10n+3} - \frac{2^2}{10n+5} - \frac{2^2}{10n+7} + \frac{1}{10n+9} \right) } \end{array}

Between 1998 and 2000, the distributed computing project PiHex used Bellard’s formula to compute the quadrillionth bit of \pi, which turned out to be… [drum roll]…

A lot of work for nothing!

No formula of this sort is known that lets you compute individual decimal digits of \pi, but it’s cool that we can do it for 1/\pi^2, at least if Almkvist and Guillera’s formula is true.

Someday I’d like to understand any one of these Ramanujan-type formulas. The search for lucid conceptual clarity that makes me love category theory runs into a big challenge when it meets the work of Ramanujan! But it’s a worthwhile challenge. I started here with one of Ramanujan’s easiest formulas:

• John Baez, Chasing the Tail of the Gaussian: Part 1 and Part 2, The n-Category Café, 28 August 28 and 3 September 2020.

But the ideas involved in this formula all predate Ramanujan. For more challenging examples one could try this paper:

• Srinivasa Ramanujan, Modular equations and approximations to \pi, Quarterly Journal of Mathematics, XLV (1914), 350–372.

Here Ramanujan gave 17 formulas for pi, without proof. A friendly-looking explanation of one is given here:

• J. M. Borwein, P. B. Borwein and D. H. Bailey, Ramanujan, modular equations, and approximations to pi or How to compute one billion digits of pi, American Mathematical Monthly 96 (1989), 201–221.

So, this is where I’ll start!

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