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Eagle River, Alaska August 2025
Fall 2025 Updates
October 5, 2025 I added to Math Notes two articles I wrote in 2021, one a review of a video on Erdos for the AMS notices Erdos 100 Review in AMS Notices 2021, and another article on my work with Cem Yildirim on the beginning of the GPY method in 1999-2001 for a Turkish High School Magazine in 2021: Turkish Magazine version , Original English manuscript.
October 1, 2025: I am giving a colloquium talk at SJSU on Oct. 15, 2025. Here is a slightly longer abstract:
Zeros of the Riemann Zeta-Function on the Critical Line
Abstract: The Riemann Hypothesis (RH) states that all non-trivial zeros of the Riemann zeta-function lie on the vertical "critical line" with real part 1/2 in the complex plane. This conjecture was made by Riemann in 1859. While the evidence that RH is true is overwhelming, no one has yet found a path towards a proof. Being more modest, one can try to prove that there are lots of zeros on the critical line. The most successful method is due to Levinson in 1974; the best result by this method obtained in 2020 is that asymptotically at least 41.7% of the zeros are on the critical line and at least 40.7% are on the critical line and also simple zeros.
A major new discovery on the Riemann zeta-function was made by Montgomery in 1973 concerning the vertical distribution of zeros, including what is called the Pair Correlation Conjecture. Montgomery assumed RH in his work, and showed that at least 2/3 of the zeros are simple. Since RH is assumed, it is impossible to prove anything about critical zeros by this pair correlation method. Starting in 2023, I have been working with 5 other mathematicians on removing RH from the pair correlation method, and we can now use this method to obtain conditional results on critical zeros. For example, assuming the Pair Correlation Conjecture but not the Riemann Hypothesis, one obtains that asymptotically 100% of the zeros are on the critical line. If we assume in place of RH that all the zeros are in a thin vertical region around the critical line, then the pair correlation method proves that at least 2/3 of the zeros must be on the critical line itself.
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:
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