The Riemann Hypothesis and Gamma Conditions

We prove that the Riemann Hypothesis holds if and only if for all 0 < λ < 12 and all 12 < σ < 1, the function e− xx− λ can be approximated, in the L1(R+, xσ − 1dx)-norm, by functions of the type ƒ(x) = Φ(x) − Φ(2x) + Φ(3x) − + ··· (x > 0), where Φ is the Laplace transform of the function χ[α,β]φ for some φ ∈ L1(R+, dx) and some 0 < α < β. We also construct a sequence { ψλn}∞n = 1 of functions of the above type and verify that for all λ > 0 and for all x > 0, ψλn(x) ↦ e−xx−λ (n ↦ ∞), and that for any ρ > 0, any λ > 0, and any real σ, we have ∫∞ρ |ψλn (x) − e−xx−λ| xσdx ↦ 0 (n ↦ ∞).