On the computation of the n^th decimal digit of various transcendental numbers

2009/12/02 by Simon Plouffe, Plouffe, Simon
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Identities #Benford’s Law and Fraud Detection

paper · pdf · doi:10.48550/arxiv.0912.0303

Abstract

A method for computing the n'th decimal digit of pi in O(n³ log(n)³) time and with very little memory is presented here. The computation is based on the recently discovered Bailey-Borwein-Plouffe algorithm and the use of a new algorithm that simply splits an ordinary fraction into its components. The algorithm can be used to compute other numbers like zeta(3), pi*sqrt(3), pi² and 2/sqrt(5) ln(phi) where phi is the golden ratio. The computation can be achieved without having to compute the preceding digits. I claim that the algorithm has a more theoretical rather than practical interest, I have not found a faster algorithm or proved that it cannot exist.

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