Let Ξ be the xi-function defined by Ξ(i z) = 1/2(z^2 - 1/4) π^(-z/2 - 1/4) Γ(1/2 z + 1/4) ζ(z + 1/2). Ξ(z/2)/8 can be viewed as the Fourier transform of the signal Φ(t) = sum_(n = 1)^∞(2π^2 n^4 e^(9t) - 3π n^2 e^(5t)) e^(-π n^2 e^(4t)) for t element R>=0. Then denote the Fourier transform of Φ(t) e^(λ t^2) as H(λ, z), ℱ_t[Φ(t) e^(λ t^2)](z) = H(λ, z). The Riemann hypothesis is equivalent to the conjecture that Λ<=0.

de Bruijn constant | xi-function