# Pi: The Next Generation

In *Pi. A source book* the editors L. Berggren, J. Borwein and P. Borwein, assembled a number of reprints that sketch the history of pi, its mathematical importance and the broad interest that it has received through the centuries from the Rhind papyrus till modern times. The last edition (3rd edition, 2004, to which I will refer as SB3) added several papers that related to the computation of the digits of pi by digital computers. Rather than extending this with more recent developments (SB3 was already some 800 pages), it was decided to collect this computational aspect in a new volume. This "*The next generation*" volume got the rightful subtitle "*a source book on the recent history of pi and its computation*". Because it extends the papers on digital computation that were added in SB3, the trailing papers of SB3 are reprised here. The papers are ordered chronologically, so of the first 14 papers in this book, 12 were already at the end of SB3.

It starts with the agm (algebraic-geometric mean) iteration attributed to Salamin and Brent who both published their papers in 1976. It generates two sequences of numbers by iteratively extending the sequences respectively with the algebraic and the geometric mean of the previous numbers. Given appropriate initial conditions, both sequences converge to a common limit related to pi. This method is widely used since these publications of 1976, but the agm idea was actually used already by Gauss and others although not in connection with computing pi. The Borwein brothers discuss a quartically convergent method based on it (1984) and Bailey and Kanada used it to compute millions of decimals of pi (1988). The number of digits computed today has exceeded these computer experiments by many orders of magnitudes and several papers in this book survey the history, and the diversity of formulas and methods and the successive records reached.

There are, besides the classical methods to compute pi, also several computational methods to generate the expansion of pi. For example, a completely different spigot algorithm computes the decimals of pi one by one but using only integer arithmetic (originally from 1995 and extended in 2006). In a more classical vein is the BBP algorithm (named after the authors Bailey, Borwein and Plouffe) which allows to compute a set of binary (or hexadecimal) digits of pi without the need to compute all the previous ones (1997). This is of course a great help when computing trillions of digits. Of course there are a a number of papers devoted to Ramanujan's notebooks with formulas to compute pi.

There are also some papers on the proof of irrationality of pi, and of related numbers such as its roots, ζ(2), ζ(3); (i.e. Apéry's constant), Catalan's constant etc. The investigation of the properties of the digits of pi, in particular the normality of pi (still unproved) is discussed and computationally tested. The tests can be nicely visualised using random walks and color coding. Normality means that every possible sequence of *m* successive digits is equally probable for any basis and for any *m*.

The papers are reprinted in their original format, thus with different fonts, lay-out, etc. It happens that the end of a previous chapter or article is still on the first page of the reprint or the start of the next one is on the last page. Even some totally unrelated announcement that appeared at the end of the original journal paper, it is reprinted here unaltered. Just as one would in a pre-digital age collect photocopies of the papers. Nevertheless, the book has an overall name and subject index, which is not obvious in this case. Since the papers come from many different journals (and even some chapters of a book) not all of these papers may be readily available or even known to an interested researcher, or in this case, it may even be a lay person who is interested. Many of the papers have authors that are the main players in the field: David Bailey, Bruce Berndt, and Jonathan and Peter Borwein. As this book was being printed one of its editors, Jonathan (Jon) Borwein, passed away on 2 August 2016. So it was probably too late to add a dedication or a note in this book. This collection he helped to compile and containing several papers that he coauthored, can be considered one of his last gift to the scientific community.

**Submitted by Adhemar Bultheel |

**23 / Aug / 2016