Calculating ζ(3) faster

A few days ago I wrote about computing ζ(3). I spent most of that post discussing simple but inefficient methods of computing ζ(3), then mentioned that there were more efficient methods.

I recently ran across a paper [1] that not only gives more efficient methods for computing ζ(3), it gives a method for generating methods. For every positive integer s, the “meta method” yields a method of computing ζ(3). As s increases, the methods get more complicated but also more efficient.

The simplest method, corresponding to s = 1 is

\zeta(3) = \frac{5}{2} \sum_{n=1}^\infty (-1)^{n-1} \frac{1}{\dbinom{2n}{n} n^3}

There are several things to say about this series. First, it goes back to Apéry and is not new in [1], but [1] puts it into the context of a sequence of series. Second, the series converges quickly because the binomial coefficient in the denominator grows exponentially, something like 4n. Third, the series alternates, which suggests it could be accelerated [2], though this may not be that important because you could derive series that converge even faster by increasing s.

The series for s = 2 is more complicated, but converges faster. It has in its denominator the binomial coefficient (2n, n) as before, but also the binomial coefficient (3n, n) which grows even faster. The series for s = 3 has the binomial coefficients (4n, n) and (3n, n) in the denominator.

How fast do these binomial coefficients grow? Let’s just say “very fast” for now. The next post will get more specific.

Related posts

[1] Faster and Faster Convergent Series for ζ(3). Available here.

[2] Series acceleration methods, like Aitken acceleration and newer methods, generally work better on alternating series.

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