We consider some of Jon Borwein's contributions to the high-precision computation of $\pi$ and the elementary functions, with particular reference to the fascinating book Pi and the AGM (Wiley, 1987) by Jon and his brother Peter Borwein.

Here "AGM" is the arithmetic-geometric mean, first studied by Euler, Gauss and Legendre. Because the AGM has second-order convergence, it can be combined with the fast multiplication algorithms of Schönhage and Strassen (1971), Fürer (2007), or Harvey, van der Hoeven and Lecerf (2014), to give fast algorithms for the $n$-bit computation of $\pi$ and more generally the elementary functions $\exp$, $\log$, $\sin$, $\tan$ etc. These algorithms run in "almost linear" time $O(M(n)\log n)$, where $M(n)$ is the time for $n$-bit multiplication.

The talk will survey a few of the results and algorithms, from the time of Archimedes to the present day, that were of interest to Jon. In several cases they were either discovered or improved by him.

As the title suggests, this lecture features mathematical identities. The identities we have chosen (we hope) are interesting, fascinating, surprising, and beautiful! We have attempted to choose identities that we think Jon would have enjoyed. In particular, $\pi$ and sums of squares will be featured. Many of the identities are due to Ramanujan. Further topics in which the identities are ensconced include partitions, continued fractions, theta functions, Bessel functions, $q$-series, other infinite series, and infinite integrals.

We study analytic properties of the Tornheim zeta function ${\mathcal W}(r,s,t)$, which is also named after Mordell and Witten. In particular, we evaluate the function ${\mathcal W}(s,s,\tau s)$ ($\tau>0$) at $s=0$ and, as our main result, find the derivative of this function at $s=0$, which turns out to be surprisingly simple. These results were first conjectured using high-precision calculations based on an identity due to Crandall that involves a free parameter and provides an analytic continuation. This identity was also the main tool in the eventual proofs of our results. Furthermore, we derive special values of a permutation sum and study an alternating analogue of ${\mathcal W}(r,s,t)$. We conclude this talk with some remarks and results on higher-order Tornheim zeta functions and their derivatives. (Joint work with Jon Borwein).

I will discuss the problem of computing the $n$-th Bernoulli number for a single large value of $n$. First I will review several existing algorithms for this problem, and give asymptotic bounds for their running time. Then I will explain the surprising observation that by combining these algorithms, we obtain a new algorithm which is more efficient by a logarithmic factor. (Joint work with Edgar Costa.)

We show that the Mahler measure of every Borwein polynomial (polynomials with coefficients in $\{-1,0,1\}$) can be expressed as the maximal Lyapunov exponent of a matrix co-cycle that arises in the spectral theory of binary substitutions of constant length. In this way, Lehmer's problem on polynomials with minimal Mahler measures becomes equivalent to a natural question from the spectral theory of binary constant-length substitutions

Let $\psi:\mathbb R_+\to\mathbb R_+$ be a non-increasing function. A real number $x$ is said to be $\psi$-Dirichlet improvable if it admits an improvement to Dirichlet's theorem, that is if the system $$|qx-p|< \psi(t) \ \ {\text{and}} \ \ |q|<t$$ has a non-trivial integer solution for all large enough $t$. In this talk, I will briefly explain that the Hausdorff measure of the set of $\psi$-Dirichlet non-improvable numbers obeys a zero-infinity law for a large class of dimension functions.

Together with the Lebesgue measure-theoretic results established by Kleinbock and Wadleigh (2016), our results contribute to building a complete metric theory for the set of Dirichlet non-improvable numbers.

(This is joint work with D. Kleinbock, N. Wadleigh and B-W. Wang.)

For all $2\leq x \leq 10^{19}$ the logarithmic integral exceeds the number of primes beneath $x$. That is $\pi(x) < \textrm{li}(x)$. Littlewood famously proved that $\pi(x) > \textrm{li}(x)$ infinitely often. In recognition of the investigations by Skewes, the first such $x$ for which this is true is sometimes called Skewes number. I shall give a survey of results on Skewes number. This is joint work with Dave Platt in Bristol.