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Register a new userThe Borwein brothers, Pi and the AGM
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Richard P. Brent
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We consider some of Jonathan and Peter Borweins contributions to the high-precision computation of $pi$ and the elementary functions, with particular reference to their book Pi and the AGM (Wiley, 1987). Here AGM is the arithmetic-geometric mean of Gauss and Legendre. Because the AGM converges quadratically, it can be combined with fast multiplication algorithms to give fast algorithms for the $n$-bit computation of $pi$, and more generally the elementary functions. These algorithms run in almost linear time $O(M(n)log n)$, where $M(n)$ is the time for $n$-bit multiplication. We outline some of the results and algorithms given in Pi and the AGM, and present some related (but new) results. In particular, we improve the published error bounds for some quadratically and quartically convergent algorithms for $pi$, such as the Gauss-Legendre algorithm. We show that an iteration of the Borwein-Borwein quartic algorithm for $pi$ is equivalent to two iterations of the Gauss-Legendre quadratic algorithm for $pi$, in the sense that they produce exactly the same sequence of approximations to $pi$ if performed using exact arithmetic.
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In this paper, we study properties of the coefficients appearing in the $q$-series expansion of $prod_{nge 1}[(1-q^n)/(1-q^{pn})]^delta$, the infinite Borwein product for an arbitrary prime $p$, raised to an arbitrary positive real power $delta$. We use the Hardy--Ramanujan--Rademacher circle method to give an asymptotic formula for the coefficients. For $p=3$ we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent $delta$ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the $p=3$ case.
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