Half-isolated Zeros and the Density Hypothesis
Many important consequences of the Riemann Hypothesis would remain true even if there were some zeros off the critical line, provided these exceptions to the Riemann Hypothesis are suitably rare. We can unconditionally prove some results on the rarity of possible exceptions, which give partial control over the distribution of primes, but the central estimates have resisted improvement for 50 years.
I'll describe a new approach to this problem based on a principle of finding DIophantine structure in such possible counterexamples, which overcomes some key obstructions to progress on applications to primes. As a consequence, if we assume that the zeros of the Zeta function lie on finitely many vertical lines then we obtain consequences for primes which are almost as strong as those implied by the Riemann Hypothesis itself.