The Riemann Hypothesis & The P vs NP Problem
Does every efficiently verifiable problem also have an efficiently solvable answer?
GitHub Repository
github.com/sdcastillo/Riemann
This presentation dives into the heart of computational complexity, optimization, and algorithm design to explore the P vs NP problem and the Riemann Zeta Hypothesis.
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What You Will Learn
Motivation & Real-World Impact
Why solving P vs NP matters and how it dictates what remains solvable under the strict physical limits of time, memory, energy, and hardware.
Intuition & NP-Completeness
A breakdown of the core difficulties in verifying versus solving.
The Riemann Hypothesis
Why the zeros of the Riemann Zeta Function govern the oscillatory behavior of prime numbers — culminating in the assertion that every non-trivial zero lies on the critical line.
Feasibility & Runtime Growth
To understand the scale of these problems, we break down algorithmic efficiency using Big O notation:
The Riemann Zeta Function
\[\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}\]
Analytic continuation and functional equation connect the behavior of primes to complex analysis.
The Riemann Hypothesis
All non-trivial zeros of \(\zeta(s)\) have real part \(\frac{1}{2}\).
Applications of Analytic Number Theory
Cryptography & Security
The distribution of prime numbers underpins RSA encryption and modern public-key cryptography. Better understanding of prime gaps directly impacts security parameters.
Quantum Computing
Connections between the Riemann zeros and quantum chaos suggest possible pathways for quantum algorithms to tackle problems in number theory and complexity classes like P vs NP.
Data Science & Large-Scale Modeling
Analytic number theory techniques appear in prime-based hashing, random number generation, and modeling of large-scale systems with irregular distributions.
Physics & Quantum Chaos
The statistical distribution of Riemann zeros mirrors energy level spacing in quantum chaotic systems, linking number theory to theoretical physics.
About the Presenter
B.S. Mathematics (honors) from UMass Amherst + actuarial science background + 6+ years building large-scale computing systems.
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