X-Nico

Generalized Riemann hypothesis

The yet to be verified proof of Harald Helfgott of this conjecture verifies the GRH for several thousand small characters up to a certain imaginary part to obtain sufficient bounds that prove the conjecture for all integers above 10²⁹, integers below which have already been verified by brute force.

Goldbach's weak conjecture

In 2012 and 2013, Peruvian mathematician Harald Helfgott released a pair of papers claiming to improve major and minor arc estimates sufficiently to unconditionally prove the weak Goldbach conjecture.

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In 2013, Harald Helfgott claimed to have fully proved the conjecture for all odd integers greater than 5 (rather than the much larger $e^\{3100\}\backslash approx\; 2\; \backslash times\; 10^\{1346\}$, implied by previous results).

see also