# Jonathan M. Borwein

# Commemorative Conference

## 25—29 September, 2017

# ◄Number Theory, Special Functions and Pi

Theme chaired by Richard Brent

**Keynote talk:**Jonathan Borwein, Pi and the AGM

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

Here "AGM" is the

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.

*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.

Identities in the Spirit of Jonathan Borwein

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.

Derivatives and fast evaluation of the Tornheim zeta function

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).

Computing Bernoulli numbers

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.)

Binary substitutions of constant length and Mahler measures of Borwein pol$

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

Improving Dirichlet's approximation theorem

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.)

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.)

A very Skewed result

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.