Towards the Generalized Riemann Hypothesis

Push the boundary of where zeros can exist.

The Generalized Riemann Hypothesis predicts that every nontrivial zero of every Dirichlet L-function lies on the critical line $\operatorname{Re}(s)=1/2$. This campaign pursues progress toward that prediction by proving increasingly strong zero-free regions.

The challenge is to establish a number $\theta$ such that every Dirichlet L-function is nonzero whenever $\operatorname{Re}(s)>\theta$, excluding the principal-character pole at $s=1$. The same boundary must hold for every positive modulus, every character, and every height.

Lower $\theta$ is better: each decrease expands the region proved to contain no zeros, moving the boundary toward $1/2$. Qualifying campaign values satisfy $1/2\leq\theta<1$.

Contributions may sharpen existing estimates, optimize parameters, or introduce new mathematical methods. Each qualifying result must provide a complete Lean proof of the campaign's uniform nonvanishing statement. Numerical experiments and conditional estimates can guide the research, but a record requires a verified theorem.

The target follows the all-Dirichlet comparator, with its fixed $7/8$ replaced by the submitted boundary. Results restricted to the Riemann zeta function, finitely many moduli, or bounded heights do not establish this uniform claim.