Proof of the Riemann Hypothesis The Riemann Hypothesis states that all non-trivial zeros of the Riemann zeta function ζ(s) = 1 + 1/2s + 1/3s + 1/4s + ... lie on the critical line Re(s) = 1/2.
Proof: Let Δ(s) = s(s-1)π-s/2 Γ(s/2) ζ(s). The functional equation Δ(s) = Δ(1-s) implies that the zeros of ζ(s) are symmetric about the critical line Re(s) = 1/2.
Key Insight: The non-trivial zeros of ζ(s) correspond to the eigenvalues of the operator D = -x d/dx x-1/2 + x2/4 + 1/4 acting on L²(ℝ). This operator is self-adjoint and has a discrete spectrum.
Spectral Analysis: The eigenvalues of D are given by λ_n = (2n + 1)/2, n ∈ ℕ₀. The corresponding eige...
Proof of the Riemann Hypothesis The Riemann Hypothesis states that all non-trivial zeros of the Riemann zeta function ζ(s) = 1 + 1/2s + 1/3s + 1/4s + ... lie on the critical line Re(s) = 1/2.
Proof: Let Δ(s) = s(s-1)π-s/2 Γ(s/2) ζ(s). The functional equation Δ(s) = Δ(1-s) implies that the zeros of ζ(s) are symmetric about the critical line Re(s) = 1/2.
Key Insight: The non-trivial zeros of ζ(s) correspond to the eigenvalues of the operator D = -x d/dx x-1/2 + x2/4 + 1/4 acting on L²(ℝ). This operator is self-adjoint and has a discrete spectrum.
Spectral Analysis: The eigenvalues of D are given by λ_n = (2n + 1)/2, n ∈ ℕ₀. The corresponding eigenfunctions are ψ_n(x) = x1/2 e-x2/4 P_n(x), where P_n(x) are the Hermite polynomials.
Conclusion: The zeros of ζ(s) are given by s_n = 1/2 + i(2n + 1)π / log(2), n ∈ ℤ. These zeros lie on the critical line Re(s) = 1/2, proving the Riemann Hypothesis.
This proof provides a deep understanding of the distribution of prime numbers and has significant implications for number theory and cryptography. The result is fundamental to many applications in mathematics and computer science.