Riemann Hypothesis Presentation Notes

Introduction

• Presentation by Alex Kontorovich, mathematician
• Discusses the Riemann Hypothesis as a key unsolved problem in mathematics.
• Prize of $1 million from the Clay Institute for a solution.
• Importance: Affects fields like cryptography and quantum physics due to its link with prime numbers.

Understanding Prime Numbers

• Prime numbers as the building blocks of whole numbers.
• Euclid's theorem: There are infinitely many primes.
• Interest in predicting where primes appear in natural numbers.

Gauss's Contributions

• Carl Friedrich Gauss: Created tables of primes, noticed patterns.
• Introduced the Prime Counting Function to track primes.
• Graph of Prime Counting Function shows jumps at prime numbers.
• Gauss's Conjecture: The proportion of primes around x is similar to 1/log(x).

Euler and the Zeta Function

• Leonard Euler: Early work on infinite series and convergence.
• Euler discovered that the limit of the series of 1/n^2 is π²/6, linking pi to squares.
• Defined the Zeta Function:
  • Zeta(s) = sum(1/n^s) for n=1 to infinity.
  • Converges when s > 1.
• Zeta function can be expressed as an infinite product over primes.

Riemann's Innovations

• Bernhard Riemann: Extended Zeta function to complex inputs.
• Introduced complex numbers: a number in the form a + bi, where i is the imaginary unit.
• Used analytic continuation to extend Zeta function to the complex plane.
• Non-trivial zeros of the Zeta function found in the critical strip (real part of s between 0 and 1).

The Riemann Hypothesis

• Riemann hypothesized that all non-trivial zeros lie on the critical line (real part of s = 1/2).
• Importance: If true, it would reveal crucial information about the distribution of prime numbers.

Connection of Zeta Function to Prime Distribution

• Modified Gauss's Prime Counting Function: Steps up by log(p) instead of 1 at each prime.
• Riemann found a wave corresponding to the Zeta function to approximate the modified prime counting function.
• Each non-trivial zero contributes a harmonic, smoothing the function.
• Proved all harmonics would match the modified prime counting function if non-trivial zeros lie on the critical line.

Conclusion

• Riemann's hypothesis links prime number distribution to the location of non-trivial Zeta zeros.
• Despite extensive computational checks (over 10 trillion zeros), no proof has been found.
• The only way to confirm the hypothesis is through rigorous mathematical proof.