The Denseness of the Closure of Some Nyman–Beurling Linear Manifolds Implies the Absence of Zeroes of Certain Combinations of Riemann Zeta-Functions in the Critical Strip

The famous Nyman–Beurling theorem states that the absence of zeroes in the Riemann zeta-function in the half-plane Res > 1/p, p > 1, is equivalent to the circumstance in which the closure of the linear manifold of the functions f(x)=∑k=1nαkϑkx, where 0<ϑk≤1, with the condition ∑k=1nakϑk=0, is dense in Lp(0,1). Here, we show that if the closure of linear manifolds of the same functions but with the conditions ∑k=1nakϑkl=0 with l = 2, 3, 4 is dense in Lp(0,1), then certain combinations of Riemann zeta-functions are free from zeroes in the half-plane Res > 1/p, p > 1—like, e.g., the function g2(s)=2s−1ζ(s−1)+ζ(s) for l = 2. Similar results for larger integer l can be established. The connections between the Riemann zeta-function, including the question concerning the location of its zeroes, with different symmetry aspects of numerous physical systems are well established, and recently they were highlighted also for supersymmetric quantum mechanics.