Absolutely continuous restrictions of a Dirac measure and non-trivial zeros of the Riemann zeta function (Q533961)

Let \(C([0,1])\) be the space of complex valued continuous function on the unit interval. The author proves that there exists an infinite dimensional subspace \(R\) of \(C([0,1])\) such that for each non-trivial zero \(s+1\) of the Riemann zeta-function, there holds \[ f(1)=\int_0^1sx^sf(x)\,dx,\qquad\forall f\in R. \] The subspace \(R\) is dense in \(L^2([0,1])\) if and only if the Riemann Hypothesis holds.