It seems that at the beginning of this year a major breakthrough on prime numbers was achieved. I learned about it from this blog: link.

Let me summarize the result quickly for you if you don’t want to read the other blog post. Almost a hundred years ago Jensen and Pólya proved that the Riemannian hypothesis is equivalent to the statement that a certain infinite family of polynomials \(J_{n,d}\) has only real roots.

In February of this year a quite short paper was posted on the arXiv:1902.07321 (meanwhile the paper is published in PNAS) proving that for each \(d\) the polynomials \(J_{n,d}\) have only real roots for almost all \(n\).