Does anyone know the current progress in showing the Riemann hypothesis? I was only able to find [this](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-20/issue-1/At-least-two-fifths-of-the-zeros-of-the-Riemann/bams/1183554910.full) paper of Conrey that says at least 40% of the zeros of the Riemann Zeta function lie on the critical line.



Current Status of the Riemann Hypothesis
In terms of fraction of zeros on the critical line (which seems to be your question), the best result to date is 41.05% by Bui, Conrey and Young ("[More than 41% of the zeros of the zeta function are on the critical line](https://arxiv.org/abs/1002.4127)" arXiv:1002.4127). Of course, this is only one measure of progress.


