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arXiv · 2004.11711

Unconditional applicability of Lehmer's measure to the two-term Machin-like formula for $π$

Sanjar M. Abrarov·Rehan Siddiqui·Rajinder K. Jagpal·Brendan M. Quine
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Abstract

Lehmer defined a measure $$ μ=\sum\limits_{j=1}^J\frac{1}{\log_{10}\left(\left|β_j\right|\right)}, $$ where the $β_j$ may be either integers or rational numbers in a Machin-like formula for $π$. When the $β_j$ are integers, Lehmer's measure can be used to determine the computational efficiency of the given Machin-like formula for $π$. However, because the computations are complicated, it is unclear if Lehmer's measure applies when one or more of the $β_j$ are rational. In this article, we develop a new algorithm for a two-term Machin-like formula for $π$ as an example of the unconditional applicability of Lehmer's measure. This approach does not involve any irrational numbers and may allow calculating $π$ rapidly by the Newton$-$Raphson iteration method for the tangent function.

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Sanjar M. Abrarov, Rehan Siddiqui, Rajinder K. Jagpal, Brendan M. Quine. 2021-06-08. Unconditional applicability of Lehmer's measure to the two-term Machin-like formula for $π$. https://doi.org/10.3888/tmj.23-2

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