 CARMA COLLOQUIUM
 Speaker: Prof. Richard Brent, CARMA, The University of Newcastle
 Title: Primes, the Riemann zetafunction, and sums over zeros
 Location: Room SR118, SR Building (and online via Zoom) (Callaghan Campus) The University of Newcastle
 Time and Date: 4:00 pm, Thu, 11^{th} Mar 2021

Join via Zoom, or join us in person (max room capacity is 9 people).
 Abstract:
First, I will give a brief introduction to the Riemann zetafunction ζ(s) and its connection with prime numbers. In particular, I will mention the famous “explicit formula” that gives an explicit connection between Chebyshev’s primecounting function ψ(x) and an infinite sum that involves the zeros of ζ(s). Using the explicit formula, many questions about prime numbers can be reduced to questions about these zeros or sums over the zeros.
Motivated by such results, in the second half of the talk I will consider sums of the form ∑φ(γ), where φ is a function satisfying mild smoothness and monotonicity conditions, and γ ranges over the ordinates of nontrivial zeros ρ = β + iγ of ζ(s), with γ restricted to be in a given interval. I will show how the numerical estimation of such sums can be accelerated, and give some numerical examples.
The new results are joint work with Dave Platt (Bristol) and Tim Trudgian (UNSW). For preprints, see arXiv:2009.05251 and arXiv:2009.13791.
 [Permanent link]
 CARMA SEMINAR
 CARMA Special Semester in Computation and Visualisation
 Speaker: Prof. Richard Brent, CARMA, The University of Newcastle
 Title: Algorithms for the Multiplication Table Problem
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Time and Date: 3:00 pm, Tue, 29^{th} May 2018
 To participate remotely, connect to the ViewMe meeting called "carmaspecial" (you can enter that name, or the meeting number 1689883675). This will be persistant for future talks in this series.
The ViewMe client is free and you do not need an account. You can install ViewMe on a computer or phone to take part, or use the web interface (Firefox or Chrome) at https://viewme.ezuce.com/webrtc/?meetingID=1689883675.
It's quite easy to use, but for assistance please contact Andrew.Danson@newcastle.edu.au. Some guides are available at https://viewme.ezuce.com/support/guidestutorials/.
 Abstract:
Let $M(n)$ be the number of distinct entries in the multiplication table for
integers smaller than $n$. More precisely, $M(n) := \{ij \mid\ 0<= i,j <n\}$. The order of magnitude of $M(n)$ was established in a series of papers by various authors, starting with Erdös (1950) and ending with Ford (2008), but an asymptotic formula for $M(n)$ is still unknown. After describing some of the history of $M(n)$ I will consider two algorithms for computing $M(n)$ exactly for moderate values of $n$, and several Monte Carlo algorithms for estimating $M(n)$ accurately for large $n$. This leads to consideration of algorithms, due to Bach (198588) and Kalai (2003), for generating random factored integers  integers $r$ that are uniformly distributed in a given interval, together with the complete prime factorisation of $r$. The talk will describe ongoing work with Carl Pomerance (Dartmouth, New Hampshire) and Jonathan Webster (Butler, Indiana).
Bio: Richard Brent is a graduate of Monash and Stanford Universities. His research interests include analysis of algorithms, computational complexity, parallel algorithms, structured linear systems, and computational number theory. He has worked at IBM Research (Yorktown Heights), Stanford, Harvard, Oxford, ANU and the University of Newcastle (NSW). In 1978 he was appointed Foundation Professor of Computer Science at ANU, and in 1983 he joined the Centre for Mathematical Analysis (also at ANU). In 1998 he moved to Oxford, returning to ANU in 2005 as an ARC Federation Fellow. He was awarded the Australian Mathematical Society Medal (1984), the Hannan Medal of the Australian Academy of Science (2005), and the Moyal Medal (2014). Brent is a Fellow of the Australian Academy of Science, the Australian Mathematical Society, the IEEE, ACM, IMA, SIAM, etc. He has supervised twenty PhD students and is the author of two books and about 270 papers. In 2011 he retired from ANU and moved to Newcastle to join CARMA, at the invitation of the late Jon Borwein.
 [Permanent link]
 CARMA SEMINAR
 Speaker: Prof. Richard Brent, CARMA, The University of Newcastle
 Title: Jonathan Borwein and Pi
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Time and Date: 1:59 pm, Wed, 14^{th} Mar 2018
 Abstract:
The late Professor Jonathan Borwein was fascinated by the constant
$\pi$. Some of his talks on this topic can be found on the CARMA website.
This homage to Jon is based on my talk at the Jonathan Borwein Commemorative
Conference. I will describe some algorithms for the highprecision
computation of $\pi$ and the elementary functions, with particular reference
to the book Pi and the AGM by Jon and his brother Peter Borwein.
Here "AGM" is the arithmeticgeometric mean
of Gauss and Legendre. Because the AGM has secondorder convergence, it
can be combined with FFTbased fast multiplication algorithms to give fast
algorithms for the \hbox{$n$bit} computation of $\pi$.
I will survey a few of the results and algorithms that were of interest to
Jon. In several cases they were either discovered or improved by him. If
time permits, I will also mention some new results that would have been of
interest to Jon.
 [Permanent link]
 CARMA SEMINAR
 Speaker: Prof. Richard Brent, CARMA, The University of Newcastle
 Title: Some Identities involving Products of Gamma Functions: a Case Study in Experimental Mathematics
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Time and Date: 2:00 pm, Tue, 27^{th} Oct 2015
 Abstract:
We consider identities satisfied by discrete analogues of Mehtalike integrals.
The integrals are related to Selberg’s integral and the Macdonald conjectures.
Our discrete analogues have the form
$$S_{\alpha,\beta,\delta} (r,n) :=
\sum_{k_1,...,k_r\in\mathbb{Z}}
\prod_{1\leq i < j\leq r}
k_i^\alpha  k_j^\alpha^\beta
\prod_{j=1}^r k_j^\delta
\binom{2n}{n+k_j},$$
where $\alpha,\beta,\delta,r,n$ are nonnegative integers subject to certain restrictions.
In the cases that we consider, it is possible to express $S_{\alpha,\beta,\delta} (r,n)$ as a
product of Gamma functions and simple functions such as powers of two.
For example, if $1 \leq r \leq n$, then
$$S_{2,2,3} (r,n) =
\prod_{j=1}^r
\frac{(2n)!j!^2}{(nj)!^2}.$$
The emphasis of the talk will be on how such identities can be obtained,
with a high degree of certainty, using numerical computation. In other cases
the existence of such identities can be ruled out, again with a high degree of
certainty. We shall not give any proofs in detail, but will outline the ideas
behind some of our proofs. These involve $q$series identities and arguments
based on nonintersecting lattice paths.
This is joint work with Christian Krattenthaler and Ole Warnaar.
 [Permanent link]
 CARMA DISCRETE MATHEMATICS INSTRUCTIONAL SEMINAR
 Speaker: Prof. Richard Brent, CARMA, The University of Newcastle
 Title: Bounds on the Hadamard maximal determinant problem using the Lovasz local lemma
 Location: Room V101, Mathematics Building (Callaghan Campus) The University of Newcastle
 Time and Date: 3:00 pm, Thu, 18^{th} Jul 2013
 Abstract:
I will explain how the probabalistic method can be used to obtain lower bounds for the Hadamard maximal determinant problem, and outline how the Lovasz local lemma (Alon and Spencer, Corollary 5.1.2) can be used to improve the lower bounds.
This is a continuation of last semester's lectures on the probabilistic method, but is intended to be selfcontained.
 [Permanent link]
 CARMA DISCRETE MATHEMATICS INSTRUCTIONAL SEMINAR
 Speaker: Prof. Richard Brent, CARMA, The University of Newcastle
 Title: The Probabilistic Method continues
 Location: Room V101, Mathematics Building (Callaghan Campus) The University of Newcastle
 Time and Date: 3:00 pm, Thu, 23^{rd} May 2013
 Abstract:
We continue on the Probabilistic Method, looking at Chapter 4 of Alon and Spencer. We will consider the second moment, the Chebyshev's inequality, Markov's inequality and Chernoff's inequality.
 [Permanent link]
