Absolutely continuous restrictions of a Dirac measure and non-trivial zeros of the Riemann zeta function

It is shown that the Dirac measure δ(f)=f(1) defined on the Banach space C([0,1]) of complex valued continuous functions defined on the interval [0,1], has an absolutely continuous restriction to an infinite dimensional subspace R of C([0,1]), that is. f(1)=10l(x)f(x)dx, ∀f∈. Each non-trivial zero of the Riemann zeta function determines a different Radon-Nikodym density l∈L1([0,1]). The Riemann Hypothesis holds if and only if none of these densities belongs to L2([0,1]) or if and only if R is dense in L2([0,1]). © 2011 Académie des sciences.