- CARMA COLLOQUIUM
- Speaker: Prof. Richard Brent, CARMA, The University of Newcastle
- 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).
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.
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