Description
Riemann Hypothesis (1859, open): All non-trivial zeros of the Riemann zeta function ζ(s) lie on the critical line Re(s) = 1/2.
Consequences if true: Best possible error term in prime-counting π(x) ∼ Li(x); many theorems in number theory rely on RH or GRH (Generalized RH for Dirichlet L-functions). Clay Millennium Prize problem.
Source: Wikipedia
Dependency Flowchart (High-Level)
graph TD
DefZeta["Def: Riemann zeta\nζ(s) = Σ n⁻ˢ, Re(s)>1"]
DefCont["Def: Analytic continuation\nζ(s) to ℂ except s=1"]
DefZero["Def: Non-trivial zero\nζ(ρ)=0, 0<Re(ρ)<1"]
DefCrit["Def: Critical line\nRe(s) = 1/2"]
ThmPNT["Thm: Prime Number Theorem\nπ(x) ∼ x/log x"]
ConjRH["Conj: Riemann Hypothesis\nAll zeros on Re(s)=1/2"]
ThmEquiv["Thm: RH ⟺ Error in π(x)\n|π(x)−Li(x)| = O(x^½ log x)"]
DefZeta --> DefCont
DefCont --> DefZero
DefZero --> ConjRH
DefCrit --> ConjRH
ThmPNT --> ThmEquiv
ConjRH --> ThmEquiv
classDef definition fill:#b197fc,color:#fff,stroke:#9775fa
classDef theorem fill:#51cf66,color:#fff,stroke:#40c057
classDef conjecture fill:#ffd43b,color:#000,stroke:#f59f00
class DefZeta,DefCont,DefZero,DefCrit definition
class ThmPNT,ThmEquiv theorem
class ConjRH conjecture