Question

Okay, let's analyze the document and then discuss progress on the problem in the last 10 years, incorporating relevant external information.

Analysis of the Document

This research paper (dated January 27, 2025 - note that it's a future date, so we should treat it as a hypothetical or preliminary version) focuses on improving lower bounds for the proportion of simple and critical zeros of the Riemann zeta function. Here's a breakdown:

• Key Problem: Determining what percentage of the nontrivial zeros of the Riemann zeta function are simple (i.e., not repeated roots) and lie on the critical line (Re(s) = 1/2).
• Background:
  • The Riemann Hypothesis (RH) is a central conjecture stating that all nontrivial zeros have a real part of 1/2.
  • Montgomery (1973), assuming RH, showed at least 2/3 of the zeros are simple.
  • Later work by Montgomery & Taylor improved the proportion to >67.25% and more recently Chirre, Gonçalves, and de Laat [CGdL20] obtained by this method 67.9%.
  • Conrey, Ghosh, and Gonek, and Bui and Heath-Brown achieved >70% assuming RH.
  • The paper [BGSTB24] removed the assumption of RH in the pair correlation method, obtaining some unconditional results.
• Main Contributions of This Paper (Hypothetical):
  • Assumes a weaker condition than RH: that zeros are in a narrow vertical box around the critical line.
  • Proves (conditionally) that at least 2/3 of the zeros are simple and at least 2/3 of the zeros are on the critical line.
  • Proves (conditionally) that at least 1/3 of the zeros are both simple and on the critical line (a result already known unconditionally).
  • Demonstrates the pair correlation method can be used to prove results on both vertical and horizontal distribution.
• Methodology:
  • Uses Montgomery's pair correlation method.
  • Generalizes Montgomery's result.
  • Corrects errors in a previous paper [BGSTB24].
  • Uses complex Fourier transforms and Tsang kernels.
• Results (Theorem 2): States precise numerical lower bounds for the proportions of simple and critical zeros under the assumption of zeros being confined to thin vertical boxes (given some small constant 'b').

Progress in the Last 10 Years (Roughly 2014 - 2024/25)

Here's a summary of progress based on the information in the paper and broader knowledge:

• Unconditional Results on the Critical Line: Significant progress has been made in improving the unconditional lower bound on the proportion of zeros lying on the critical line. The best result as of November 2023, is due to Pratt, Robles, Zaharescu, and Zeindler [PRZZ20], who proved that more than 41.7% of the zeros are on the critical line, and also that more than 40.7% of the zeros are on the critical line and simple.

• Simple Zeros (Conditional): Assuming the Riemann Hypothesis, the best result as of November 2023, is due to Conrey, Ghosh, and Gonek [CGG98] showed on RH and an additional hypothesis that at least 19/27 = 70.3703% of the zeros are simple, and later Bui and Heath-Brown [BHB13] showed that this result holds on RH alone.

• Pair Correlation Method: The document itself suggests that the method continues to be refined and applied, with advancements in removing the RH assumption and exploring the interplay between vertical and horizontal distributions of zeros. The document's authors ([BGSTB24] and this paper) are actively contributing to this line of research.

• Other Techniques: Beyond the pair correlation method, other techniques from analytic number theory are being used to attack the problem, though the precise details may be highly technical and outside the scope of a general overview.

Key Takeaways Regarding Progress

1. Unconditional progress is slow but steady: Improvements to the unconditional proportion of zeros known to be on the critical line are hard-won and involve sophisticated techniques.
2. The Riemann Hypothesis remains a major barrier: Assuming RH allows for much stronger results, but proving RH remains one of the greatest unsolved problems in mathematics.
3. The pair correlation method is a powerful tool: The research presented suggests that even without assuming RH, this method has the potential to provide new insights.
4. Refinements and Variations: Research focuses on refining existing methods (like pair correlation) with weaker assumptions and variations of previously known approaches.

In Conclusion

While significant progress has been made towards proving the Riemann Hypothesis, it remains one of the greatest unsolved problems in mathematics. Significant progress has also been made towards improving the known proportion of zeros on the critical line. The results from the paper [BGSTB24] and others provide insight into potential avenues for exploration.