The Riemann Hypothesis: Past, Present and a Letter Through Time

TL;DR

This paper surveys 165 years of research on the Riemann Hypothesis, introduces a novel historical-inspired method for approximating zeros, and discusses potential avenues for future proof strategies.

Contribution

It presents an original 'Letter to Riemann' method using classical mathematics to approximate zeros and proves these approximations lie on the critical line, offering new insights into the hypothesis.

Findings

Approximate zeros with high accuracy using primes less than 13.

Proven that these approximations lie exactly on the critical line.

Developed a geometric perspective connecting trace formulas and Euler products.

Abstract

This paper, commissioned as a survey of the Riemann Hypothesis, provides a comprehensive overview of 165 years of mathematical approaches to this fundamental problem, while introducing a new perspective that emerged during its preparation. The paper begins with a detailed description of what we know about the Riemann zeta function and its zeros, followed by an extensive survey of mathematical theories developed in pursuit of RH -- from classical analytic approaches to modern geometric and physical methods. We also discuss several equivalent formulations of the hypothesis. Within this survey framework, we present an original contribution in the form of a "Letter to Riemann," using only mathematics available in his time. This letter reveals a method inspired by Riemann's own approach to the conformal mapping theorem: by extremizing a quadratic form (restriction of Weil's quadratic form in modern language), we obtain remarkable approximations to the zeros of zeta. Using only primes less than 13, this optimization procedure yields approximations to the first 50 zeros with accuracies ranging from $2.6 \times 10^{-55}$ to $10^{-3}$. Moreover we prove a general result that these approximating values lie exactly on the critical line. Following the letter, we explain the underlying mathematics in modern terms, including the description of a deep connection of the Weil quadratic form with the world of information theory. The final sections develop a geometric perspective using trace formulas, outlining a potential proof strategy based on establishing convergence of zeros from finite to infinite Euler products. While completing the commissioned survey, these new results suggest a promising direction for future research on Riemann's conjecture.