CARMA COLLOQUIUM Speaker: Prof. Richard Brent, Australian National University Title: Primes, the Riemann zeta-function, 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, 11th 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 zeta-function 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 prime-counting 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, Australian National University 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, 29th 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/guides-tutorials/. 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