The research map

Zero free regions and explicit bounds

Riemann Hypothesis

Widen the region where zeta provably has no zeros, and make every constant explicit.

Explicit formula and prime counting error terms

Riemann Hypothesis

Turn a zero free region into a usable bound on the prime counting error.

Random matrix theory and pair correlation

Riemann Hypothesis

The statistics of the zeros look like the statistics of eigenvalues of large random unitary matrices.

Montgomery pair correlation conjecture

Riemann Hypothesis

A precise form for the distribution of gaps between consecutive zeros.

Verified computation of zeros at large height

Riemann Hypothesis

Compute zeros high up the critical line with certificates a third party can re-check.

Formalization in a proof assistant

Riemann Hypothesis

Barriers: relativization, natural proofs, algebrization

P versus NP

Circuit lower bounds

P versus NP

Lower bounds against constant depth circuits with counting gates

P versus NP

Proof complexity

P versus NP

Geometric complexity theory

P versus NP

Density and almost all results

Collatz Conjecture

Stopping time statistics

Collatz Conjecture

Exhaustive verification with reproducible certificates

Collatz Conjecture

Formalization of the statement and known partial results

Collatz Conjecture

Circle method: major and minor arcs

Goldbach Conjecture

Leverage from the ternary case

Goldbach Conjecture

Exhaustive verification and the exceptional set

Goldbach Conjecture