In a paper on arXiv Simon Plouffe gives the formula

which he derives from an equation in Abramowitz and Stegun (A&S).

It took a little while for me to understand what Plouffe intended. I don’t mean my remarks here to be criticism of the author but rather helpful hints for anyone else who might have difficulty reading the paper.

He says “Here the *B*_{n} are positive and *n* is even.” This doesn’t mean that the *B*_{n} *are* positive; it means that we should *make* them positive if necessary by taking the absolute value.

Plouffe says that he derives his estimate from an equation on page 809 of A&S, but at least in my copy of A&S, there are no equations on page 809. I believe he had in mind equation 23.2.16, which is on page 807 in my edition.

This equation says

The first inequality in the paper does not appear directly in A&S but can be derived from the equation above. The approximation for π at the top of the post approximates ζ(2*n*) by 1. You could obtain a better approximation for π by using a better approximation for ζ(2*n*), which Plouffe does, retaining the first few terms in Euler’s product representation for the zeta function.

How good is the approximation above? Plotting the approximation error on a log scale gives nearly a straight line, i.e. the error decreases exponentially.

This plot was produced with the following Mathematica code.

f[m_] := (2 Factorial[2m] / Abs[BernoulliB[2m] 2^(2m)])^(1/(2m)) ListPlot[Table[Log[Abs[f[m] - Pi]], {m, 1, 20}]]

## Related posts

- Euler product for sine
- Five posts on computing pi
- Special numbers (Bell, Bernoulli, Stirling, etc.)
- Why Euler’s product can’t be used to find zeros of zeta

My version of A&S (AMS 55) also shows 23.2.16 on page 807.

Thanks. I didn’t think the book had changed since 1964, but I suppose there are minor differences in printings.

In any case, equation numbers are better than page numbers.