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

Between 1998 and 2000, the distributed computing project PiHex used Bellard’s formula to compute the quadrillionth bit of 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 but it’s cool that we can do it for 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:

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So you think the fans of put the in the wrong place? That’s an interesting conjecture as far as groupoid cardinality goes: it’s hard to find a nice groupoid of cardinality or but easier to find one of cardinality

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I wonder if is more fundamental than , in some sense, e.g. in that it appears in zeta function values.

So you think the fans of put the in the wrong place? That’s an interesting conjecture as far as groupoid cardinality goes: it’s hard to find a nice groupoid of cardinality or but easier to find one of cardinality

One can argue that the really important number is

Is there some definition that leads to complex groupoid cardinalities?

Not that I know; the closest thing I know is Fiore and Leinster’s paper Objects of categories as complex numbers, which is really quite nice.