Mathematics & Applied Mathematics - Riemann Hypothesis
Problem Statement: Prove or disprove the Riemann Hypothesis, which states that all non-trivial zeros of the Riemann zeta function have real part equal to 1/2.
Why This Exemplifies the Field: This 160+ year old problem is considered the most important unsolved problem in mathematics, with profound implications for prime number theory and connections to physics, making it the archetypal example of deep mathematical inquiry.
Evaluation Criteria:
Complete mathematical proof published in peer-reviewed journal(s)
Verification by multiple independent mathematical review committees
Proof accepted by consensus among experts in number theory and analysis
Clarification of connections to prime number theory
If proven true, classification of all exceptions if any exist
Broad implications for related conjectures and number theory clearly established
Feasibility Assessment: Extremely challenging, potentially requiring 10-30 years. Despite over 160 years of attempts, a proof remains elusive. Likely requires novel connections between different branches of mathematics. Recent advancements in analytical techniques and computational verification provide some optimism.
Impact on the Field: Would resolve one of the most significant open problems in mathematics with profound implications for prime number theory, cryptography, and quantum chaos. Would enable new classes of mathematical proofs that rely on the hypothesis and provide deeper understanding of the distribution of prime numbers.
