September 14, 2025

I am excited about my joint work with 5 other mathematicians in two papers that were placed on the arXiv earlier this year. Both of them present new insights into the zeros of the Riemann zeta-function and introduce a conditional method to prove that some of the zeros lie on the critical line, which, of course, is directly connected to the Riemann Hypothesis. These papers are relatively simple, the results are easy to state, and the proofs are short.  These are also the main reasons given by referees for the quick rejections both of these papers have been receiving from journals.  Why is this?  I don't know,  but I can easily explain why I think these papers solve an interesting problem. Suppose the zeros of the zeta function are your enemy, and they will do everything in their power to stay off of the critical line.  The best result concerning this scenario is the 2020 result of Kyle Pratt, Nicolas Robles, Alexandru Zaharescu, and Dirk Zeindler that uses Levinson's method to prove more than 41.7% of the zeros must be on the critical line.  The problem we consider is: Can we do better if we first assume an additional property of the zeros not directly connected to being on the critical line?  In the first paper we assume the zeros are all forced to be very close to the critical line;  the distance being within C/log t at height t.  What we prove is that, using Montgomery's pair correlation method, if you take C=3/20 then you can push at least 2/3 of the zeros onto the critical line. In fact, if C=0.0005, you can push 67.25% of the zeros onto the critical line, which is close to the limit of the method. In the second paper, a related pair correlation method proves that asymptotically 100% of zeros are on the critical line. Here, instead of assuming the horizontal distance from the critical line is restricted, we assume Montgomery's Pair Correlation Conjecture, which provides an asymptotic formula for the vertical distribution of pairs of zeros. While the Pair Correlation Conjecture is likely to be very difficult to prove in its own right, it says nothing about the horizontal position of zeros, and up to now, no one suspected it could be used to push zeros horizontally onto the critical line.  For more details, see the papers:  

1. Pair Correlation of Zeros of the Riemann Zeta Function I: Proportions of Simple Zeros and Critical Zeros, by Siegfred Alan C. Baluyot, Daniel Alan Goldston, Ade Irma Suriajaya, Caroline L. Turnage-Butterbaugh

2. Pair Correlation Conjecture for the Zeros of the Riemann Zeta-function I: Simple and Critical Zeros, by Daniel A. Goldston, Junghun Lee, Jordan Schettler, Ade Irma Suriajaya