I know that it has been proven that $\mathrm{Re}(\rho)<1$, and RH is $\mathrm{Re}(\rho)=1/2$, where $\rho$ is zero of Riemann zeta function. Can someone summarize, or link me to very recent improvement on this "Real part of zero"?

Specifically, I have few questions related to this.

  1. Is real part of zero of zeta function always rational?

  2. What is best bound for the real part? That is, what is the smallest $\delta$ discovered, that for all zero of zeta function, following always holds. $$|\mathrm{Re}(\rho)-1/2|<\delta $$

  1. That's not known.
  2. The best known bound for the non-trivial zeros is $\delta=\dfrac12$.

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