# Jonathan M. Borwein

# Commemorative Conference

## 25—29 September, 2017

# ◄Experimental Mathematics and Visualisation►

Theme chaired by David Bailey

**Keynote talk:**Computer discovery and analysis of large Poisson polynomials

In earlier studies of lattice sums arising from the Poisson equation of mathematical physics, it was established that when $x$ and $y$ are
rational, the lattice sum
$$\frac{1}{\pi} \sum_{m,n \text{ odd}} \frac{\cos(m\ \pi\ x) \times cos(n\ \pi\ y)}{m^2 + n^2}$$
is always equal to the logarithm of an algebraic number $A$ that depends on $x$ and $y$. We were also able to find explicit minimal
polynomials associated with $A$ for a few specific rational arguments $x$ and $y$. Based on these results, Kimberley conjectured a
number-theoretic formula for the degree of $A$ in the case $x = y = 1/s$ for some integer $s$. These earlier studies were hampered by the
enormous cost and complexity of the requisite computations. In this study, we addressed the Poisson polynomial problem with
significantly more capable computational tools, and were able to confirm Kimberley's formula for all integers $s$ up to 52 (except for
$s = 41, 43, 47, 49, 51$, which are still too costly to test), and also for $s = 60$ and $s = 64$. As far as we are aware, these computations,
which employed up to 64,000-digit precision, producing polynomials with degrees up to 512 and integer coefficients up to $10^{229}$,
constitute the largest successful integer relation computations performed to date. By examining the computed results, we found
connections to a sequence of polynomials defined in a 2010 paper by Savin and Quarfoot. These investigations subsequently led to a proof
of Kimberley's formula and also the fact that when $s$ is even, the polynomial is palindromic.

Gamma and Factorial in the Monthly

Since its inception in the nineteenth century, the American Mathematical Monthly has published over fifty papers on the Gamma function or equivalently the factorial
function. Over half of these were on Stirling's formula. We survey these papers, which include a Chauvenet prize winning paper by Philip J. Davis and a paper by the
Fields medallist Manjul Bhargava, and highlight some features in common. We also identify some surprising gaps and attempt to fill them, especially on the "inverse Gamma
function".

This is joint work with the late Jonathan M. Borwein.

This is joint work with the late Jonathan M. Borwein.

New Ramanujan Mock Theta Function Identities

In his last letter to Hardy, Ramanujan defined ten mock theta functions of order 5 and three of order 7. He stated that the three mock theta functions of order 7 are
not related. We show how we discovered and proved new identities for the fifth order functions and that there are actually surprising relationships between the order 7
functions.

On the Computation of Mathematical Constants and the Implications of HPC Trends

In preparation for Pi Day celebrations at the University of Technology Sydney (UTS) in 2013, David Bailey, Jon Borwein, Glenn Wightwick and the
presenter engaged in an exercise to compute far-flung digits of $\pi^2$ and Catalan's constant utilising BBP formulae for each. A brief review will be
presented of this effort and the resulting paper, which was awarded the 2017 Levi L Conant Prize by the American Mathematical Society. Recent trends in
High-Performance Computing (HPC) will be explored, as they have implications for similar experimental mathematics computations in the future.

Phase Portraits of Hyperbolic Geometry

Phase plotting is a useful way of illustrating the behavior of functions on the complex plane. Exploiting natural connections between the complex plane and hyperbolic geometry,
we introduce a similar method of plotting for differential geometry by assigning unique colors to the preimages of geodesics. Interestingly, for some representations the portraits are also phase portraits in the classical sense. This work is collaboration with Paul Vrbik.

Sums of Palindromes: An Approach via Automata

We prove, using a decision procedure based on automata, that every natural number is the sum of at most 9 natural numbers whose base-2 representation is a palindrome. We consider
some other variations on this problem, and prove similar results. We argue that heavily case-based proofs are a good signal that a decision procedure may help to automate the proof.

How to turn a fractal inside out and discover theorems

I will describe recent results, and the role that visualization played, in the discovery of a simple method for turning an attractor of a graph directed iterated function system inside out.
The resulting approach to tiling theory provides insight into very sophisticated theorems of others concerning aperiodic tilings. This is joint work with A. Vince and A. Grant. An article on the topic will appear in the December issue of the Math Monthly.

A magnetic double integral

In a recent study of how the output voltage of a Hall plate is affected by the shape of the plate and the size of its contacts, Udo Ausserlechner has come up with a
remarkable double integral that can be viewed as a generalization of the classical elliptic "AGM" integral. In my talk I will discuss transformation properties of the
integral, which were experimentally observed by Ausserlechner, as well as its analytical and arithmetic features including connections to the AGM and other special
functions.

This is joint work with David Broadhurst (Open University, UK).

This is joint work with David Broadhurst (Open University, UK).