Lecture Notes: The Riemann Hypothesis

Introduction

• Speaker: Alex Kantorovich, Professor of Mathematics
• Topic: Riemann Hypothesis - a major unsolved problem in mathematics
• Significance: Part of the Millennium Problems by the Clay Institute, with a $1 million prize for solving it
• Importance: Vital for understanding the distribution of prime numbers; impacts fields like cryptography and quantum physics

Understanding the Riemann Hypothesis

• Prime Numbers: Building blocks of all numbers, appearing seemingly randomly
• Gauss's Contribution:
  • Developed the prime counting function, showing where primes appear
  • Found similarity between the prime counting function and the logarithmic integral function
  • Conjectured the distribution of primes could be predicted

Historical Context

• Leonard Euler's Contribution:
  • Studied infinite series and their convergence
  • Developed the zeta function to study sums of series
  • Discovered the zeta function's link to prime numbers

Bernard Riemann's Breakthrough

• Complex Analysis:
  • Pioneered by Riemann, extended the zeta function to complex numbers
  • Used analytic continuation to expand the domain of the zeta function
• Zeta Zeros:
  • Points where the zeta function equals zero
  • Non-trivial zeros are critical and lie in the 'critical strip' where the real part is between 0 and 1
  • Riemann Hypothesis: All non-trivial zeros lie on the 'critical line' where the real part is 1/2

Implications of the Hypothesis

• Connection to Prime Numbers:
  • Zeta zeros are connected to the distribution of prime numbers
  • If true, it would predict the distribution of primes perfectly
• Current Status:
  • Unproven, despite computational checks confirming pattern up to billions of zeros
  • Requires rigorous mathematical proof to be resolved

Conclusion

• Importance of Proof:
  • Highlighted need for mathematical proof over computational checks
  • The hypothesis remains a central challenge in mathematics due to its profound implications

Additional Notes

• Historical Figures Mentioned:
  • Carl Friedrich Gauss, Euclid of Alexandria, Leonard Euler
• **Mathematical Concepts: **
  • Infinite series, complex numbers, logarithms
  • Analytic continuation and convergence/divergence of series

The lecture explores the fascinating mystery of the Riemann Hypothesis, its historical development, and its profound significance in understanding the distribution of prime numbers.