Explicit methods in number theory: computation, theory and application

This Proposal will unify three problems in number theory: primitive roots, Diophantine quintuples, and linear independence of zeroes of the Riemann zeta-function. This unification will be achieved using state-of-the-art explicit estimates, in which the Future Fellow is an established expert. This Proposal will establish new results on primitive roots, and apply these to signal processing, cryptography, and cybersecurity. A powerful mix of computational and analytic number theory will reduce the quintuples problem to a soluble level. Pursuing relations between the zeta zeroes will overhaul many current results. Research in these problems is very active. Striking advances and applications are now possible and achievable with this Proposal.