A generalization of Beurling's criterion for the Riemann hypothesis

It is known that the Riemann hypothesis holds if and only if the function χ(0,1)χ(0,1) can be approximated by linear combinations of uαuα in L2(0,1)L2(0,1). Here uα(x)uα(x) is defined by [α/x]−α[1/x][α/x]−α[1/x] for 0<α<10<α<1. In this note we generalize the Beurling's equivalent condition by replacing the function χ(0,1)χ(0,1) with χ(a,b)χ(a,b) for any 0≤a