16:00
Abstract. I will talk about projects in which we combine heuristics with computational data to develop a theory in problems where it was previously hard to be confident of the guesses that there are in the literature.
1/ "Speculations about the number of primes in fast growing sequences". Starting from studying the distribution of primes in sequences like $2^n-3$, Jon Grantham and I have been developing a heuristic to guess at the frequency of prime values in arbitrary linear recurrence sequences in the integers, backed by calculations.
If there is enough time I will then talk about:
2/ "The spectrum of the $k$th roots of unity for $k>2$, and beyond". There are many questions in analytic number theory which revolve around the "spectrum", the possible mean values of multiplicative functions supported on the $k$th roots of unity. Twenty years ago Soundararajan and I determined the spectrum when $k=2$, and gave some weak partial results for $k>2$, the various complex spectra. Kevin Church and I have been tweaking MATLAB's package on differential delay equations to help us to develop a heuristic theory of these spectra for $k>2$, allowing us to (reasonably?) guess at the answers to some of the central questions.